[A-wal (OP)]

I'll start a new topic for this because I don't want to hijack someone else's.

If two observers are circling each other without the distance between them changing then of course they would each see the other's watch ticking at the same rate as their own watch but how could following a path that spirals inwards cause a different result than a direct path?

Easiest case: From PoV of Earth, the remote clock is getting closer (blueshift) but is moving (redshift).  If you adjust the angle just right, the two cancel.  The relativistic effect is a function of speed, but the Doppler effect is a function of the rate of reduction of separation.  A circular path would be red shifted (all dilation), but a direct path is dominated by Doppler.  Somewhere between is a balance.  The math isn't too hard to work out.

Shouldn't time dilation also be a function of the rate of change of separation?

Imagine they are the only two objects in the universe. Any spiral path is now meaningless because there's no point of reference to create a spiral.

Earth is the reference.  Remember, rotation is still absolute, not relative.
So no need for a 3rd object for it to be a spiral.

Again though, if there were only two objects in the universe then wouldn't spin be entirely relative?

If we look at it from Object A's frame, B is orbitting around us (tidally locked). If we then switch to Object B's frame, we are stationary and A is spinning.

Now you could say that if Object B is orbiting Object A then Object B is under constant acceleration but if Object A is spinning then there is no acceleration.

What if there were only one object, could it spin and/or accelerate? I don't see how. Is acceleration really frame independent or is it just as relative as velocity?

[Halc]

Shouldn't time dilation also be a function of the rate of change of separation?

No.  If Alice is orbiting Bob at high speed, Alice's clock goes much slower than Bob's, despite no change in separation ever.  But there is no Doppler effect noticed by either observer because that is a function of change in distance.

Again though, if there were only two objects in the universe then wouldn't spin be entirely relative?

If Earth was the sole existing thing (one object), we still could tell that it spins because of its oblate shape, and the fact that it is easier to put an object into orbit going east than it is going west.  Not sure if there'd be an obvious east and west though without stars or something to trek across the sky.  Interesting question: How do you tell which way your planet spins without looking up?  Coriolis effect of course.  The storms would twirl the other way if the spin was westward.  There'd be no hurricanes if there was no spin at all.

Now you could say that if Object B is orbiting Object A then Object B is under constant acceleration but if Object A is spinning then there is no acceleration.

If object B is accelerating, then so it A, spinning or not. Conservation of momentum demands this. So does Newton's 3rd law of motion.

What if there were only one object, could it spin and/or accelerate?

Both spin and acceleration are absolute.  Velocity is not.

Where does this thread get into time dilation with curved paths?  It was an interesting question, how Alice can get back home to Bob without Bob seeing her age fast or slow.

[A-wal (OP)]

Shouldn't time dilation also be a function of the rate of change of separation?

No.  If Alice is orbiting Bob at high speed, Alice's clock goes much slower than Bob's, despite no change in separation ever.  But there is no Doppler effect noticed by either observer because that is a function of change in distance.

Alice would be constantly accelerating, that's a different cause of time dilation than relative velocity. In this case Alice's watch is running slower than Bob's from Alice's and Bob's frame. So yes, you could use acceleration to add to the time dilation caused by relative velocity and cancel out blue shift to cause a watch that's moving towards you tick at the normal rate.

Time dilation caused by relative velocity is purely a function of the rate of change of separation so the question becomes whether or not you need other objects to accelerate yourself. I think acceleration can be defined as an alteration in the rate of change of separation between objects, a description of acceleration that would require it to be relative to something else.

Again though, if there were only two objects in the universe then wouldn't spin be entirely relative?

If Earth was the sole existing thing (one object), we still could tell that it spins because of its oblate shape, and the fact that it is easier to put an object into orbit going east than it is going west.  Not sure if there'd be an obvious east and west though without stars or something to trek across the sky.  Interesting question: How do you tell which way your planet spins without looking up?  Coriolis effect of course.  The storms would twirl the other way if the spin was westward.  There'd be no hurricanes if there was no spin at all.

I would dispute that the Earth would have that shape and that it would make any difference which direction you put an object into orbit if there were no other objects in the universe. I'd rephrase that last sentence as: There could be no hurricanes if Earth were an isolated object.

Now you could say that if Object B is orbiting Object A then Object B is under constant acceleration but if Object A is spinning then there is no acceleration.

If object B is accelerating, then so it A, spinning or not. Conservation of momentum demands this. So does Newton's 3rd law of motion.

But isn't that circular reasoning, by defining momentum within a coordinate system that doesn't apply or even make sense if there's only two objects in the system.

What if there were only one object, could it spin and/or accelerate?

Both spin and acceleration are absolute.  Velocity is not.

I don't see it. I'm not trying to be argumentative, I'm glad you don't agree because it means we can have an interesting debate. This is my favourite physics question.

Where does this thread get into time dilation with curved paths?  It was an interesting question, how Alice can get back home to Bob without Bob seeing her age fast or slow.

It was supposed to, I got sidetracked. I think this is more interesting anyway and it is connected. In the spirit of getting sidetracked, I had a conversation about this ages ago and was told about Newton's Bucket, could a bucket filled with water in an otherwise empty universe spin, could the water raise at the sides? I thought it couldn't and was told I believed in Mach's Principle. When I looked it up I was like YES, this is exactly how I see it.

Mach's Principle isn't well defined but I think it can be summed up as: Acceleration is relative. You can outrun light when you accelerate, there's a point behind you past which no light will ever reach you (Rindler Horizon) and the distance between you and that horizon decreases if you increase your acceleration. The distance decreases at a slower rate with the same acceleration increase the higher our starting acceleration, that distance never reaches zero.

Now if we shine our headlights to look the light in front moving away from us that light will always be moving away from slower than c. We can reduce that light's velocity relative to us by increasing our acceleration and the acceleration addition formula describing the forward light's outward velocity going from c with no acceleration to 0 with infinite acceleration I think is the same as the velocity addition formula.

[Janus]

Mach's Principle isn't well defined but I think it can be summed up as: Acceleration is relative. You can outrun light when you accelerate, there's a point behind you past which no light will ever reach you (Rindler Horizon) and the distance between you and that horizon decreases if you increase your acceleration. The distance decreases at a slower rate with the same acceleration increase the higher our starting acceleration, that distance never reaches zero.

Now if we shine our headlights to look the light in front moving away from us that light will always be moving away from slower than c.

No.  The proper or "local" speed of light that you would measure would be still c.  What would change would be the coordinate speed of light as you look behind you. The coordinate speed would decrease while looking back toward the Rindler Horizon (to zero at the horizon).   It is like a distant observer looking at a black hole.  The coordinate speed of light is zero at the event horizon, but he measures the local speed of light as being c, even if that light originated near the event horizon.   

We can reduce that light's velocity relative to us by increasing our acceleration and the acceleration addition formula describing the forward light's outward velocity going from c with no acceleration to 0 with infinite acceleration I think is the same as the velocity addition formula.

You can only change the coordinate speed of light for points separated from you along the direction of your acceleration, but the local instantaneous speed of light you measure will still be c.

[A-wal (OP)]

Mach's Principle isn't well defined but I think it can be summed up as: Acceleration is relative. You can outrun light when you accelerate, there's a point behind you past which no light will ever reach you (Rindler Horizon) and the distance between you and that horizon decreases if you increase your acceleration. The distance decreases at a slower rate with the same acceleration increase the higher our starting acceleration, that distance never reaches zero.

Now if we shine our headlights to look the light in front moving away from us that light will always be moving away from slower than c.

No.  The proper or "local" speed of light that you would measure would be still c.  What would change would be the coordinate speed of light as you look behind you. The coordinate speed would decrease while looking back toward the Rindler Horizon (to zero at the horizon).   It is like a distant observer looking at a black hole.  The coordinate speed of light is zero at the event horizon, but he measures the local speed of light as being c, even if that light originated near the event horizon.

We can reduce that light's velocity relative to us by increasing our acceleration and the acceleration addition formula describing the forward light's outward velocity going from c with no acceleration to 0 with infinite acceleration I think is the same as the velocity addition formula.

You can only change the coordinate speed of light for points separated from you along the direction of your acceleration, but the local instantaneous speed of light you measure will still be c.

Yes the local speed of light will always be c, I didn't explain that very well.

If you take into account the fact that you're time dilated due to acceleration and work out how fast the light in front of you would be moving away from you if you weren't time dilated then it's velocity will be < c and and the acceleration addition formula describing the light's velocity going from c with no acceleration to 0 with infinite acceleration I think is the same as the velocity addition formula.

[Halc]

If Alice is orbiting Bob at high speed, Alice's clock goes much slower than Bob's, despite no change in separation ever.  But there is no Doppler effect noticed by either observer because that is a function of change in distance.

Alice would be constantly accelerating, that's a different cause of time dilation than relative velocity.

Indeed.  If you discount acceleration, then the only inertial solutions to Bob seeing Alices's clock run continuously at a normal rate is if Alice is stationary relative to Bob.  If she passes by, it will appear to change rates as she goes by (like the siren speeding by) and will be the normal rate for only a moment.

So yes, you could use acceleration to add to the time dilation caused by relative velocity and cancel out blue shift to cause a watch that's moving towards you tick at the normal rate.

Actually, I tried to find a solution to that and failed.  Suppose a clock is inertial and coming at me at .6c.  What can I do to make it appear to me like it is running at full speed?  Acceleration doesn't work.  I need to match its speed, instantly.  I need to be stationary relative to it.  No games of acceleration will work.  So while Alice might be able to accelerate and keep her clock running normally as seen by Bob, she isn't going to see the same on Bob's clock.  It will be substantially blue shifted.  She'd have to spiral away to get that, so there is a solution for an inertial receding clock, but not an incoming one.

Time dilation caused by relative velocity is purely a function of the rate of change of separation

No, not at all.  In an inertial frame, it is a function of speed.  I gave an example above, and the only way Alice can maintain constant separation at some nonzero speed is to accelerate around a circular path.  Her clock is dilated exactly as a function of whatever her speed is, and her acceleration (large or small) makes no difference to that.

I think acceleration can be defined as an alteration in the rate of change of separation between objects, a description of acceleration that would require it to be relative to something else.

It is already defined as a change in velocity over time (dx/dt).  It can't be something else. It isn't a relative thing, just like rotation is not relative.

If Earth was the sole existing thing (one object), we still could tell that it spins because of its oblate shape, and the fact that it is easier to put an object into orbit going east than it is going west.  Not sure if there'd be an obvious east and west though without stars or something to trek across the sky.  Interesting question: How do you tell which way your planet spins without looking up?  Coriolis effect of course.  The storms would twirl the other way if the spin was westward.  There'd be no hurricanes if there was no spin at all.

I would dispute that the Earth would have that shape and that it would make any difference which direction you put an object into orbit if there were no other objects in the universe.

The oblate shape of say Jupiter isn't caused by other objects. Are you asserting that it is, that Jupiter would be completely spherical if you took away all the stuff not Jupiter?  That thing rotates fast.
I see you've looked up Newton's bucket, which is an early empirical argument for rotation being absolute.

There could be no hurricanes if Earth were an isolated object.

If we still heated it with lamps (something to replace the sunlight), then there would very much be hurricanes as long as Earth still rotated.  If it rotated fast enough, it would fly apart, and a lone object in the sky would not be required for it.

Earth is not a point mass.  Anything that isn't a point mass has extension, and thus is made of components which can rotate about each other, so effectively, Earth is a collection of small objects which yes, rotate about each other.  A point mass is what cannot meaningfully rotate, even if there were other objects around it to provide some kind of better reference.  A point mass cannot have an oblate shape, or any other kind of shape for that matter.

If object B is accelerating, then so is A, spinning or not. Conservation of momentum demands this. So does Newton's 3rd law of motion.

But isn't that circular reasoning, by defining momentum within a coordinate system that doesn't apply or even make sense if there's only two objects in the system.

I think so, yes.  A coordinate system becomes less and less defined as you remove objects.  With two objects, only one axis remains. Momentum is pretty meaningless. With one object, distance and probably even time become meaningless. Our universe does not lack for references that do not move back and forth as the moon tugs the Earth this way and that.

What if there were only one object, could it spin and/or accelerate?

Both spin and acceleration are absolute.  Velocity is not.

I don't see it. I'm not trying to be argumentative, I'm glad you don't agree because it means we can have an interesting debate. This is my favourite physics question.

Per my argument above, if the thing has extension, then it isn't really one object, but rather this side and that side, which is two objects.  Being two objects, it can spin.
Considering a point object with no other references, I agree that spin and acceleration are completely meaningless. Even mass and force is meaningless in such a situation.

Mach's Principle isn't well defined but I think it can be summed up as: Acceleration is relative.

It seems to say quite the opposite.

You can outrun light when you accelerate, there's a point behind you past which no light will ever reach you (Rindler Horizon) and the distance between you and that horizon decreases if you increase your acceleration.

Yes. That you can't do that standing still means there is something different about accelerating than not accelerating, and that suggests acceleration is absolute.

Now if we shine our headlights to look the light in front moving away from us that light will always be moving away from slower than c.

Not in the frame of the accelerating guy with the light.  That's like shining a light straight up into the sky.  It speeds up, not slows down. You're working it out apparently in some other frame, but I don't know which one.
Shine it behind you if you want it to slow down in your coordinate frame.  It would be like shining light into a black hole (said Rindler horizon) where it sort of just freezes there, at least in your frame.

[Janus]

Mach's Principle isn't well defined but I think it can be summed up as: Acceleration is relative. You can outrun light when you accelerate, there's a point behind you past which no light will ever reach you (Rindler Horizon) and the distance between you and that horizon decreases if you increase your acceleration. The distance decreases at a slower rate with the same acceleration increase the higher our starting acceleration, that distance never reaches zero.

Now if we shine our headlights to look the light in front moving away from us that light will always be moving away from slower than c.

No.  The proper or "local" speed of light that you would measure would be still c.  What would change would be the coordinate speed of light as you look behind you. The coordinate speed would decrease while looking back toward the Rindler Horizon (to zero at the horizon).   It is like a distant observer looking at a black hole.  The coordinate speed of light is zero at the event horizon, but he measures the local speed of light as being c, even if that light originated near the event horizon.

We can reduce that light's velocity relative to us by increasing our acceleration and the acceleration addition formula describing the forward light's outward velocity going from c with no acceleration to 0 with infinite acceleration I think is the same as the velocity addition formula.

You can only change the coordinate speed of light for points separated from you along the direction of your acceleration, but the local instantaneous speed of light you measure will still be c.

Yes the local speed of light will always be c, I didn't explain that very well.

If you take into account the fact that you're time dilated due to acceleration and work out how fast the light in front of you would be moving away from you if you weren't time dilated then it's velocity will be < c and and the acceleration addition formula describing the light's velocity going from c with no acceleration to 0 with infinite acceleration I think is the same as the velocity addition formula.

You never take into account that "you" are time dilated.  Time dilation is what you measure occurring to clocks due to their relative motion with respect to you, their position relative to you in a gravitational field, or their position relative to your measured acceleration, if you are under acceleration (regardless of whether or not those clocks share your acceleration).

[A-wal (OP)]

If Alice is orbiting Bob at high speed, Alice's clock goes much slower than Bob's, despite no change in separation ever.  But there is no Doppler effect noticed by either observer because that is a function of change in distance.

Alice would be constantly accelerating, that's a different cause of time dilation than relative velocity.

Indeed.  If you discount acceleration, then the only inertial solutions to Bob seeing Alices's clock run continuously at a normal rate is if Alice is stationary relative to Bob.  If she passes by, it will appear to change rates as she goes by (like the siren speeding by) and will be the normal rate for only a moment.

And yet if you have the same situation with gravitational orbit then Alice is entirely inertial. No, that can wait. I'm not going to get sidetracked again.

So yes, you could use acceleration to add to the time dilation caused by relative velocity and cancel out blue shift to cause a watch that's moving towards you tick at the normal rate.

Actually, I tried to find a solution to that and failed.  Suppose a clock is inertial and coming at me at .6c.  What can I do to make it appear to me like it is running at full speed?  Acceleration doesn't work.  I need to match its speed, instantly.  I need to be stationary relative to it.  No games of acceleration will work.

You could run really fast in a tight circle. Technically Alice's clock will never be ticking at the same rate as yours  (at least not for any extended length of time) but you could make it tick at the same rate as your watch on average over the circle. Science is fun.

So while Alice might be able to accelerate and keep her clock running normally as seen by Bob, she isn't going to see the same on Bob's clock.  It will be substantially blue shifted.  She'd have to spiral away to get that, so there is a solution for an inertial receding clock, but not an incoming one.

But if they were the only two objects in the universe then she wouldn't be able to follow a spiral path.

Time dilation caused by relative velocity is purely a function of the rate of change of separation

No, not at all.  In an inertial frame, it is a function of speed.  I gave an example above, and the only way Alice can maintain constant separation at some nonzero speed is to accelerate around a circular path.  Her clock is dilated exactly as a function of whatever her speed is, and her acceleration (large or small) makes no difference to that.

But her clock in that situation is dilated because she is constantly accelerating, she has no velocity relative to Bob. In a two body system relative velocity is indistinguishable from the rate of change of separation. To follow a curved (accelerated) path requires at least one other object.

I think acceleration can be defined as an alteration in the rate of change of separation between objects, a description of acceleration that would require it to be relative to something else.

It is already defined as a change in velocity over time (dx/dt).  It can't be something else. It isn't a relative thing, just like rotation is not relative.

Yes but is that an accurate definition considering an object circling around you is constantly accelerating without changing their relative velocity.

If Earth was the sole existing thing (one object), we still could tell that it spins because of its oblate shape, and the fact that it is easier to put an object into orbit going east than it is going west.  Not sure if there'd be an obvious east and west though without stars or something to trek across the sky.  Interesting question: How do you tell which way your planet spins without looking up?  Coriolis effect of course.  The storms would twirl the other way if the spin was westward.  There'd be no hurricanes if there was no spin at all.

I would dispute that the Earth would have that shape and that it would make any difference which direction you put an object into orbit if there were no other objects in the universe.

The oblate shape of say Jupiter isn't caused by other objects. Are you asserting that it is, that Jupiter would be completely spherical if you took away all the stuff not Jupiter?

I am.

That thing rotates fast.

Don't care how fast it is, it's still only able to spin relative to everything that's not Jupiter.

I see you've looked up Newton's bucket, which is an early empirical argument for rotation being absolute.

And an early argument for it being relative. If the water rises up around the sides then acceleration is absolute, not going to happen.

There could be no hurricanes if Earth were an isolated object.

If we still heated it with lamps (something to replace the sunlight), then there would very much be hurricanes as long as Earth still rotated.  If it rotated fast enough, it would fly apart, and a lone object in the sky would not be required for it.

Earth is not a point mass.  Anything that isn't a point mass has extension, and thus is made of components which can rotate about each other, so effectively, Earth is a collection of small objects which yes, rotate about each other.  A point mass is what cannot meaningfully rotate, even if there were other objects around it to provide some kind of better reference.  A point mass cannot have an oblate shape, or any other kind of shape for that matter.

But if Earth were the only object in the universe then we could treat it as a whole as a point mass. The universe as a whole can't rotate, it needs a wider context for rotation to be defined.

If object B is accelerating, then so is A, spinning or not. Conservation of momentum demands this. So does Newton's 3rd law of motion.

But isn't that circular reasoning, by defining momentum within a coordinate system that doesn't apply or even make sense if there's only two objects in the system.

I think so, yes.  A coordinate system becomes less and less defined as you remove objects.  With two objects, only one axis remains. Momentum is pretty meaningless. With one object, distance and probably even time become meaningless. Our universe does not lack for references that do not move back and forth as the moon tugs the Earth this way and that.

Yes with two objects you only have one spatial dimension and with three objects you only have two spatial dimensions, a triangle. You need at least four objects to have three spatial dimensions.

Relative velocity, acceleration and spin all become meaningless in a one object universe, regardless of the size of that object. The universe as a whole can't have a spin or acceleration just as can't have a relative velocity because all are relative.

What if there were only one object, could it spin and/or accelerate?

Both spin and acceleration are absolute.  Velocity is not.

I don't see it. I'm not trying to be argumentative, I'm glad you don't agree because it means we can have an interesting debate. This is my favourite physics question.

Per my argument above, if the thing has extension, then it isn't really one object, but rather this side and that side, which is two objects.  Being two objects, it can spin.

I don't see how, this side and that side are motionless relative to each other. If you want it to spin you need a separate object, in fact you need two if you want its spin to be distinct from the other object circling it.

Considering a point object with no other references, I agree that spin and acceleration are completely meaningless. Even mass and force is meaningless in such a situation.

And that's what you get if you treat any extended object in isolation.

Mach's Principle isn't well defined but I think it can be summed up as: Acceleration is relative.

It seems to say quite the opposite.

How so? Here's a better definition of it: Any group of objects treated collectively require a separate point of reference to define any form of motion.

You can outrun light when you accelerate, there's a point behind you past which no light will ever reach you (Rindler Horizon) and the distance between you and that horizon decreases if you increase your acceleration.

Yes. That you can't do that standing still means there is something different about accelerating than not accelerating, and that suggests acceleration is absolute.

Something else must have emitted that light. I suppose if you wanted a universe with one massive object and lots of light flying around then the acceleration of that massive object would have an effect on when according to their clock the light reaches them, but with no other massive objects how could they check? Their clock would only be measuring the interior of that object and when light beams impact them with no way to define the velocity of themselves or the light.

Now if we shine our headlights to look the light in front moving away from us that light will always be moving away from slower than c.

Not in the frame of the accelerating guy with the light.  That's like shining a light straight up into the sky.  It speeds up, not slows down. You're working it out apparently in some other frame, but I don't know which one.

A light pointed up speeds up? It would still be c locally.

I'm using a frame in which the accelerating object is always stationary and taking into account that light appears to have gained forward velocity due to time dilation and length contraction. I just said time dilation before but both would obviously need to be taken into account.

Shine it behind you if you want it to slow down in your coordinate frame.  It would be like shining light into a black hole (said Rindler horizon) where it sort of just freezes there, at least in your frame.

Yes, it would continue to slow down without ever freezing. But again, c locally. That Rindler horizon gets closer to you as you increase your acceleration and a greater increase in acceleration would be needed for it to close the gap by the same amount as acceleration increases.

But there's another horizon in front of you, the light you shine ahead of you moves away from you at a velocity that approaches zero at infinite acceleration at the point where you catch up to this horizon in front of you and the Rindler horizon catches up to you, the point where the two horizon meet.

These horizons are what I think follow the velocity addition formula, with the increase in acceleration need to close the gap between the horizon and the accelerating observer being the equivalent of the increase in relative velocity need to reach the speed of light relative to another object. It would be a nice equivalence if true, acceleration is to energy as velocity is to matter.

That horizon in front of you is unreachable because that would require infinite acceleration and time in the rest of the universe speeds up from your perspective at an ever increasing rate approaching infinity if it were to approach you at a continuous rate. What does could this horizon be equivalent to?

You never take into account that "you" are time dilated.  Time dilation is what you measure occurring to clocks due to their relative motion with respect to you, their position relative to you in a gravitational field, or their position relative to your measured acceleration, if you are under acceleration (regardless of whether or not those clocks share your acceleration).

It doesn't make sense to take your own time dilation into account when it's due to relative velocity because it depends on an arbitrary choice of coordinate system but when it's due to acceleration you can put yourself at rest and go from there, judging your own time dilation by comparing your clock to inertial clocks. You'd have to take into account time dilation caused by relative velocity which would be changing due to their changing relative velocities.

Hopefully I'm describing the Rindler coordinate system, just put a horizon in front of the accelerating object always at the same distance as the Rindler horizon is behind them and that's the horizon I'm getting at. Light moves away from the accelerating object at c locally but immediately slows at an ever increasing rate as it approaches the horizon that prevent it from ever actually reaching the horizon to become frozen.

Let's call it the exponential velocity equilibrium null threshold horizon, or event horizon for short.


Sidetracked again. Anyway, Newton's bucket. It might seem ridiculous to think that one object spinning is physically equivalent to every other object spinning while following a curved path around you with a velocity proportional to it's distance but is that really much more absurd than thinking that you moving to your right is physically equivalent to everything that isn't you moving to your left, and we know that's true.

[Halc]

And yet if you have the same situation with gravitational orbit then Alice is entirely inertial.

Freefall and inertial are different things.

No, that can wait. I'm not going to get sidetracked again.

Yes, let's stick to SR for the moment.  Alice is on a string being spun about Bob.

Suppose a clock is inertial and coming at me at .6c.  What can I do to make it appear to me like it is running at full speed? ..  No games of acceleration will work.

You could run really fast in a tight circle. Technically Alice's clock will never be ticking at the same rate as yours  (at least not for any extended length of time) but you could make it tick at the same rate as your watch on average over the circle. Science is fun.

That would make it worse. To a stationary Bob, her clock appears to run at 2x his own rate.  If he slows his own clock by running in tight circles, his own clock will slow down, and hers will appear even faster in comparison to it.  No, we're trying to get them to match here.

But if they were the only two objects in the universe then she wouldn't be able to follow a spiral path.

We've been over this. Rotation inside a box is the same empirical thing as rotation without the box. If it makes you happy, put a bunch of stars way out there. I'm in this thread for the title topic: What two observers can do so that at least one of them sees the clock on of the other run at the same pace as his own.
Open a topic in New Theories if you want to push the view that either rotation or acceleration are not absolute. It doesn't belong here.

But her clock in that situation is dilated because she is constantly accelerating, she has no velocity relative to Bob.

Very wrong twice. Dilation relative to a stationary observer, and discounting gravity, is entirely a function of speed and not acceleration.  The age of the twin in the twin experiment can be expressed as a function of integrating his speed, and acceleration has nothing to do with it.  Dilation relative to a non-inertial observer is a function of acceleration, but we're expressing Alice's orbiting clock relative to Bob here.
Secondly, she very much has a velocity relative to Bob.  The ISS is moving at some 7.6 km/sec which is hardly no velocity relative to our Bob, who happens to be at one of the poles to eliminate any acceleration from Earth spin.  Velocity is change in position over time. That's the definition, not change in separation over time.

A light pointed up speeds up? It would still be c locally.

Coordinate speed, yes.

But there's another horizon in front of you, the light you shine ahead of you moves away from you at a velocity that approaches zero at infinite acceleration

You just agreed in the post above that it does the opposite:

I'm using a frame in which the accelerating object is always stationary and taking into account that light appears to have gained forward velocity due to time dilation and length contraction.

Anyway, there is no horizon in front of you.

[A-wal (OP)]

I seem to have annoyed you somehow. I'm sorry, that wasn't my intent.

And yet if you have the same situation with gravitational orbit then Alice is entirely inertial.

Freefall and inertial are different things.

I agree, but they're often treated as the same.

Suppose a clock is inertial and coming at me at .6c.  What can I do to make it appear to me like it is running at full speed? ..  No games of acceleration will work.

You could run really fast in a tight circle. Technically Alice's clock will never be ticking at the same rate as yours  (at least not for any extended length of time) but you could make it tick at the same rate as your watch on average over the circle. Science is fun.

That would make it worse. To a stationary Bob, her clock appears to run at 2x his own rate.  If he slows his own clock by running in tight circles, his own clock will slow down, and hers will appear even faster in comparison to it.  No, we're trying to get them to match here.

Oh right, if it's coming towards you then it's blue shifted so running in a circle will just make the other watch more blue shifted, that's for an observer who's moving away from you. I misread that paragraph, you meant what can you do to see them ticking at the same rate if the object is moving towards you. Nothing because the other observer has to spiral.

You should be able to have either observer seeing their watch and the other observer's watch ticking at the same rate whether they're moving towards or away from each other, just never both observers seeing both watches ticking at the same rate. One of them needs to follow a spiral path to slow down their watch from the perspective of both observers.

If the two observers are moving away from each other then the observer who is spiralling can see the watch of the observer who isn't spiralling ticking at the same rate as their own watch. If the two observers are moving towards each other then the observer who isn't spiralling can see the watch of the observer who is spiralling ticking at the same rate as their own watch.

But if they were the only two objects in the universe then she wouldn't be able to follow a spiral path.

We've been over this. Rotation inside a box is the same empirical thing as rotation without the box. If it makes you happy, put a bunch of stars way out there. I'm in this thread for the title topic: What two observers can do so that at least one of them sees the clock on of the other run at the same pace as his own.
Open a topic in New Theories if you want to push the view that either rotation or acceleration are not absolute. It doesn't belong here.

Mach's principle isn't a new theory. You said:

A coordinate system becomes less and less defined as you remove objects.  With two objects, only one axis remains. Momentum is pretty meaningless. With one object, distance and probably even time become meaningless. Our universe does not lack for references that do not move back and forth as the moon tugs the Earth this way and that.

How could you have spin in a universe with only one object? That's taking the coordinate system as a whole and rotating it, nothing changes. I don't see how you could have spin without having an external point of reference for that spin. It's a shame you don't want to talk about this, it's fascinating. Do I really have to put it in new theories if I want to make a topic for it?

But her clock in that situation is dilated because she is constantly accelerating, she has no velocity relative to Bob.

Very wrong twice. Dilation relative to a stationary observer, and discounting gravity, is entirely a function of speed and not acceleration.  The age of the twin in the twin experiment can be expressed as a function of integrating his speed, and acceleration has nothing to do with it.  Dilation relative to a non-inertial observer is a function of acceleration, but we're expressing Alice's orbiting clock relative to Bob here.

Did you mean dilation relative to a non-stationary observer? I'm not used to using relative velocity alone to solve the age discrepancy in the twin paradox and don't quite see how that could work, if you ignore acceleration you have only mirrored time dilation with no age difference at the end.

If you're going the route of using the difference in elapsed proper time on their watches to show that the Earth twin was time dilated for less time on the traveler's watch than the traveler's watch was time dilated for on the Earth twin's watch then you've already taken acceleration into account.

If you're treating the Earth bound watch as stationary and the other watch as moving then that doesn't take into account that the inertial motion was relative. On it's own it would be absolute motion, so the only way to not view the age difference at the end as a result of acceleration is to use absolute velocity.

Maybe you're not using either of those methods and I'm just not seeing how you're doing it, but even then it wouldn't invalidate the other methods that do describe the difference in age as a function of acceleration, you seem to be implying that it would and therefore gives a circling object an absolute velocity.

I like to use acceleration alone to explain the difference. Each clock is running slower than the other while they're inertial so ignore that, ignore Doppler shift because it cancels itself out, the difference in age is caused by the Earth twin's watch running fast during the acceleration because you're changing frames.

I know that acceleration itself isn't responsible because the amount of elapsed time on the watches at the end can vary using the same accelerations or you can change the acceleration and keep the same difference in elapsed time but I don't see how you can leave out the difference being caused by only one of them changing frames.

Secondly, she very much has a velocity relative to Bob.  The ISS is moving at some 7.6 km/sec which is hardly no velocity relative to our Bob, who happens to be at one of the poles to eliminate any acceleration from Earth spin.  Velocity is change in position over time. That's the definition, not change in separation over time.

But that would be an absolute velocity relative to Bob. They have no relative velocity because without a third object it's indistinguishable from Bob spinning. Even with a third object it's indistinguishable from that third object spinning while alternating moving backwards and forwards and Bob circling her.

But there's another horizon in front of you, the light you shine ahead of you moves away from you at a velocity that approaches zero at infinite acceleration

You just agreed in the post above that it does the opposite:

I'm using a frame in which the accelerating object is always stationary and taking into account that light appears to have gained forward velocity due to time dilation and length contraction.

I meant that the accelerating object is time dilated and length contracted which increases the velocity of the light moving out in front of them but they still measure it to be c so it's lost forward velocity once you take time dilation and length contraction into account. This is the opposite effect of the Rindler horizon behind them.

Anyway, there is no horizon in front of you.

The light in front of an accelerating observer wouldn't be unaffected by their acceleration any more than the light behind them would. There should be an equivalent to the Rindler horizon in front of them.

Because it's equivalent we can use the Rindler horizon anyway. Am I right in saying that the reduction in distance between the Rindler horizon and the accelerating observer follows the velocity addition formula?


Sorry if I'm getting on your nerves.

[Halc]

Oh right, if it's coming towards you then it's blue shifted so running in a circle will just make it more the other watch more blue shifted, that's for an observer who's moving away from you.

Yes, but it works from the other perspective.  If the incoming inertial observer watches an earth clock in the centrifuge, it could be seen as running at the same rate.

One of them needs to follow a spiral path to slow down their watch from the perspective of both observers.

One cannot slow down ones own clock no matter what speed, path, or acceleration is going on. A local clock measures proper time, which by definition is never dilated.

If the two observers are moving away from each other then the observer who is spiralling can see the watch of the observer who isn't spiralling ticking at the same rate as their own watch. If the two observers are moving towards each other then the observer who isn't spiralling can see the watch of the observer who is spiralling ticking at the same rate as their own watch.

Agree.  We're talking about a logarithmic spiral here and not something like an Archimedes spiral.

Open a topic in New Theories if you want to push the view that either rotation or acceleration are not absolute. It doesn't belong here.

Mach's principle isn't a new theory.
 ...
It's a shame you don't want to talk about this, it's fascinating. Do I really have to put it in new theories if I want to make a topic for it?

Mach's principle is accepted physics. What you're suggesting (spin not being absolute, and acceleration for that matter) is not, and thus doesn't belong in a thread opened in the Physics/Astronomy section of this forum. It will cause this topic to be moved there if that's what you intend to discuss here.

Dilation relative to a stationary observer, and discounting gravity, is entirely a function of speed and not acceleration.  The age of the twin in the twin experiment can be expressed as a function of integrating his speed, and acceleration has nothing to do with it.  Dilation relative to a non-inertial observer is a function of acceleration, but we're expressing Alice's orbiting clock relative to Bob here.

Did you mean dilation relative to a non-stationary observer?

Any non-accelerating thing is stationary relative to itself.  Even an accelerating thing is stationary relative to itself maybe, if the concept of stationary is defined at all in an accelerating frame.  It seems to have little meaning there. So no, I meant what I said.

I'm not used to using relative velocity alone to solve the age discrepancy in the twin paradox and don't quite see how that could work, if you ignore acceleration you have only mirrored time dilation with no age difference at the end.

The traveler's age relative to Earth can be determined solely as a function of his speed relative to Earth, but only because Earth is presumed inertial.  The same cannot be said of the traveler because he's not inertial, so the computation of Earth's clock relative to him is not just a function of Earth's speed.
I can generalize even further:  Only relative to a given arbitrary inertial frame, the dilation of a moving thing is purely a function of the moving thing's speed.

If you're going the route of using the difference in elapsed proper time on their watches to show that the Earth twin was time dilated for less time on the traveler's watch than the traveler's watch was time dilated for on the Earth twin's watch then you've already taken acceleration into account.

Traveler moves say .6c both out and back.  His clock thus runs at 80% the pace of an Earth clock, and will be 80% slow when compared.  All that can be computed knowing only the .6c and not the acceleration at all.  Maybe he stays close by and orbits furiously the whole time at that speed.  Maybe he takes a triangular path repeatedly, or spells his name in the cosmic snow.  Point is, if he does it at .6c the whole time, he'll be 20% younger when he returns.  All those cases have very different accelerations, but it matter not at all to the end result.

If you're treating the Earth bound watch as stationary and the other watch as moving then that doesn't take into account that the inertial motion was relative.

You get the same answer in any inertial frame.  It actually doesn't require Earth or anything else to be stationary. It only requires all speeds to be considered in one arbitrary frame. You get the same answer regardless of frame choice.  Choosing the frame there the beginning and end comparisons are the same point in space just makes the mathematics somewhat less complicated.

Maybe you're not using either of those methods and I'm just not seeing how you're doing it, but even then it wouldn't invalidate the other methods that do describe the difference in age as a function of acceleration, you seem to be implying that it would and therefore gives a circling object an absolute velocity.
...
I like to use acceleration alone to explain the difference.

OK.  I accelerate at 1000g for a year, according to the clock sitting in a building in Brazil.  How much time does my clock say has elapsed?  The question is easily answered if it is known that I move at 0.6c (relative to Brazil) for that day.

I know that acceleration itself isn't responsible because the amount of elapsed time on the watches at the end can vary using the same accelerations or you can change the acceleration and keep the same difference in elapsed time but I don't see how you can leave out the difference being caused by only one of them changing frames.

I didn't compute anything related to change of frames. My method is simply to pick a frame (any frame) and stick with it.

But that would be an absolute velocity relative to Bob.

You don't see the contradiction in this statement?

I meant that the accelerating object is time dilated and length contracted which increases the velocity of the light moving out in front of them but they still measure it to be c so it's lost forward velocity once you take time dilation and length contraction into account.

As Janus said, one is never dilated relative to ones self.  If I'm accelerating north, and I shine a pulse of light north, an hour later than light pulse will be more than a light hour away in my frame at the time, that is, the frame in which I am momentarily inertial.  There is no length contraction because a thing is always stationary in its own frame.

An event horizon is a threshold in space beyond which no event can ever have a causal effect on you.  There is no such horizon in front of an accelerating object in the flat spacetime of the special relativity case.  Any event that happens in front of you (say a signal sent at you) will get to you, all the faster because you're accelerating towards it instead of just waiting for the light to make the trip.

[A-wal (OP)]

Oh right, if it's coming towards you then it's blue shifted so running in a circle will just make it more the other watch more blue shifted, that's for an observer who's moving away from you.

Yes, but it works from the other perspective.  If the incoming inertial observer watches an earth clock in the centrifuge, it could be seen as running at the same rate.

Yes the non-spiralling observer can see the spiralling observer's watch ticking at the same rate as their own watch if they're moving towards each other and the spiralling observer can see the non-spiralling observer's watch ticking at the same rate as their own watch if they're moving away from each other.

If you ignore Doppler shift and only take into account time dilation the Earth observer could run really fast in a circle to make both watches tick at the same rate from their own perspective.

One of them needs to follow a spiral path to slow down their watch from the perspective of both observers.

One cannot slow down ones own clock no matter what speed, path, or acceleration is going on. A local clock measures proper time, which by definition is never dilated.

Of course, I'm surprised you thought that's what I meant. One of them needs to follow a spiral path to slow down their watch from the perspective of both observers, which the spiralling observer will see as the other watch speeding up, because they obviously can't see their own watch slowing down.

If the two observers are moving away from each other then the observer who is spiralling can see the watch of the observer who isn't spiralling ticking at the same rate as their own watch. If the two observers are moving towards each other then the observer who isn't spiralling can see the watch of the observer who is spiralling ticking at the same rate as their own watch.

Agree.  We're talking about a logarithmic spiral here and not something like an Archimedes spiral.

I don't know the difference. Maybe if I show my own ignorance you'll not feel the need to try to highlight it.

Open a topic in New Theories if you want to push the view that either rotation or acceleration are not absolute. It doesn't belong here.

Mach's principle isn't a new theory.
 ...
It's a shame you don't want to talk about this, it's fascinating. Do I really have to put it in new theories if I want to make a topic for it?

Mach's principle is accepted physics. What you're suggesting (spin not being absolute, and acceleration for that matter) is not, and thus doesn't belong in a thread opened in the Physics/Astronomy section of this forum. It will cause this topic to be moved there if that's what you intend to discuss here.

Spin and acceleration not being absolute is Mach's principle. I thought Mach's principle was neither accepted or rejected, it's an open question. From wikipedia:

The broad notion that "mass there influences inertia here" has been expressed in several forms. Hermann Bondi and Joseph Samuel have listed eleven distinct statements that can be called Mach principles, labelled Mach0 through Mach10. Though their list is not necessarily exhaustive, it does give a flavor for the variety possible.

Mach0: The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames.
Mach1: Newton's gravitational constant G is a dynamical field.
Mach2: An isolated body in otherwise empty space has no inertia.
Mach3: Local inertial frames are affected by the cosmic motion and distribution of matter.
Mach4: The universe is spatially closed.
Mach5: The total energy, angular and linear momentum of the universe are zero.
Mach6: Inertial mass is affected by the global distribution of matter.
Mach7: If you take away all matter, there is no more space.
Mach8: 'Horseshoe, def above =, 4, pi symbol, funny p, G, T, squared' is a definite number, of order unity, where funny p is the mean density of matter in the universe, and T is the Hubble time.
Mach9: The theory contains no absolute elements.
Mach10: Overall rigid rotations and translations of a system are unobservable.

Mine do work:
Acceleration is relative.
Any group of objects treated collectively require a separate point of reference to define any form of motion.

I wrote this one just before I looked that up, it's basically Mach10: Motions of a coordinate system as a whole are meaningless, having no effect on any objects within the system.

I'm not used to using relative velocity alone to solve the age discrepancy in the twin paradox and don't quite see how that could work, if you ignore acceleration you have only mirrored time dilation with no age difference at the end.

The traveler's age relative to Earth can be determined solely as a function of his speed relative to Earth, but only because Earth is presumed inertial.  The same cannot be said of the traveler because he's not inertial, so the computation of Earth's clock relative to him is not just a function of Earth's speed.

Yes, so if you want to explain their difference in age at the you can't ignore the fact that one of them accelerated and one didn't. That's not a complete picture using only relative velocity.

Very wrong twice. Dilation relative to a stationary observer, and discounting gravity, is entirely a function of speed and not acceleration.  The age of the twin in the twin experiment can be expressed as a function of integrating his speed, and acceleration has nothing to do with it.  Dilation relative to a non-inertial observer is a function of acceleration, but we're expressing Alice's orbiting clock relative to Bob here.

I can generalize even further:  Only relative to a given arbitrary inertial frame, the dilation of a moving thing is purely a function of the moving thing's speed.

An inertial frame is one without acceleration so you've written into your own definition that the age difference is caused by the fact that one accelerated and one didn't.

If you're going the route of using the difference in elapsed proper time on their watches to show that the Earth twin was time dilated for less time on the traveler's watch than the traveler's watch was time dilated for on the Earth twin's watch then you've already taken acceleration into account.

Traveler moves say .6c both out and back.  His clock thus runs at 80% the pace of an Earth clock, and will be 80% slow when compared.  All that can be computed knowing only the .6c and not the acceleration at all.  Maybe he stays close by and orbits furiously the whole time at that speed.  Maybe he takes a triangular path repeatedly, or spells his name in the cosmic snow.  Point is, if he does it at .6c the whole time, he'll be 20% younger when he returns.  All those cases have very different accelerations, but it matter not at all to the end result.

I said not more than a few lines later:

I know that acceleration itself isn't responsible because the amount of elapsed time on the watches at the end can vary using the same accelerations or you can change the acceleration and keep the same difference in elapsed time but I don't see how you can leave out the difference being caused by only one of them changing frames.

If you're treating the Earth bound watch as stationary and the other watch as moving then that doesn't take into account that the inertial motion was relative.

You get the same answer in any inertial frame.  It actually doesn't require Earth or anything else to be stationary. It only requires all speeds to be considered in one arbitrary frame. You get the same answer regardless of frame choice.  Choosing the frame there the beginning and end comparisons are the same point in space just makes the mathematics somewhat less complicated.

What? You only get that answer in the frame in which Earth is stationary. Both twins will have aged less in any other inertial frame, although the difference on their watches will of course be the same at the end in any frame, not just inertial ones.

You were treating the Earth bound watch as stationary and the other watch as moving.

Maybe you're not using either of those methods and I'm just not seeing how you're doing it, but even then it wouldn't invalidate the other methods that do describe the difference in age as a function of acceleration, you seem to be implying that it would and therefore gives a circling object an absolute velocity.
...
I like to use acceleration alone to explain the difference.

OK.  I accelerate at 1000g for a year, according to the clock sitting in a building in Brazil.  How much time does my clock say has elapsed?  The question is easily answered if it is known that I move at 0.6c (relative to Brazil) for that day.

And where's the rest of that paragraph that puts that first sentence into context?

I like to use acceleration alone to explain the difference. Each clock is running slower than the other while they're inertial so ignore that, ignore Doppler shift because it cancels itself out, the difference in age is caused by the Earth twin's watch running fast during the acceleration because you're changing frames.

I know that acceleration itself isn't responsible because the amount of elapsed time on the watches at the end can vary using the same accelerations or you can change the acceleration and keep the same difference in elapsed time but I don't see how you can leave out the difference being caused by only one of them changing frames.

I didn't compute anything related to change of frames. My method is simply to pick a frame (any frame) and stick with it.

Your method is incomplete and can't explain the difference in age without implying a preferred frame.

Secondly, she very much has a velocity relative to Bob.  The ISS is moving at some 7.6 km/sec which is hardly no velocity relative to our Bob, who happens to be at one of the poles to eliminate any acceleration from Earth spin.  Velocity is change in position over time. That's the definition, not change in separation over time.

But that would be an absolute velocity relative to Bob.

You don't see the contradiction in this statement?

Yes I do, that's my point. Alice's velocity relative to Bob is purely coordinate dependant, because there's no change in separation over time.

And where's the rest of that paragraph that puts that first sentence into context?

But that would be an absolute velocity relative to Bob. They have no relative velocity because without a third object it's indistinguishable from Bob spinning. Even with a third object it's indistinguishable from that third object spinning while alternating moving backwards and forwards and Bob circling her.

You still seem to be in combat mode. Your last two posts are very disingenuous, deliberately taking what I say out of context in attempts to obscure the point. This is an example of what you're doing:

Dilation relative to a stationary observer, and discounting gravity, is entirely a function of speed and not acceleration.  The age of the twin in the twin experiment can be expressed as a function of integrating his speed, and acceleration has nothing to do with it.  Dilation relative to a non-inertial observer is a function of acceleration, but we're expressing Alice's orbiting clock relative to Bob here.

Did you mean dilation relative to a non-stationary observer?

Any non-accelerating thing is stationary relative to itself.  Even an accelerating thing is stationary relative to itself maybe, if the concept of stationary is defined at all in an accelerating frame.  It seems to have little meaning there. So no, I meant what I said.

But you said dilation relative to a stationary observer. There would be no dilation relative to a stationary observer if:

Secondly, she very much has a velocity relative to Bob.  The ISS is moving at some 7.6 km/sec which is hardly no velocity relative to our Bob, who happens to be at one of the poles to eliminate any acceleration from Earth spin.  Velocity is change in position over time. That's the definition, not change in separation over time.

Time dilation occurs when objects are in motion relative to each other, stationary objects are never time dilated in special relativity. Some other condescending sentence that implies you don't already know all this. You're free to use any inertial frame any object that's at at rest in this frame will see clocks that are in motion in frame as time dilated.

So you meant dilation relative to an observer who is stationary relative to themselves?

I don't know what I've done to offend you but I very formally and sincerely apologise if anything I said in any of my previous posts came across as rude or disrespectful. I hope we can continue this discussion in the mannor in which we started it.

I meant that the accelerating object is time dilated and length contracted which increases the velocity of the light moving out in front of them but they still measure it to be c so it's lost forward velocity once you take time dilation and length contraction into account.

As Janus said, one is never dilated relative to ones self.  If I'm accelerating north, and I shine a pulse of light north, an hour later than light pulse will be more than a light hour away in my frame at the time, that is, the frame in which I am momentarily inertial.  There is no length contraction because a thing is always stationary in its own frame.

Sigh. An accelerating observer takes into account the fact that they are time dilated and length contracted because of their acceleration. They can indirectly measure this by virtue of the fact that they can measure their own acceleration.

They then work out what the speed of light relative to themselves moving away in front of them would be if they were not time dilated and length contracted and they can see that the speed of light relative to themselves moving away in front of them slows as they increase their acceleration, with the reduction in speed of light relative to themselves moving away in front of them lessening with the same increase in acceleration as their overall acceleration increases in such a way that the speed of light relative to themselves moving away in front of them approaches without ever reaching 0. I think this follows the velocity addition formula.

An event horizon is a threshold in space beyond which no event can ever have a causal effect on you.  There is no such horizon in front of an accelerating object in the flat spacetime of the special relativity case.  Any event that happens in front of you (say a signal sent at you) will get to you, all the faster because you're accelerating towards it instead of just waiting for the light to make the trip.

I'm taking about a signal moving away from you. It should mirror the Rindler horizon. So does the distance decrease between the Rindler horizon and the accelerating observer as they increase their acceleration match the velocity addition formula?

[Halc]

We're talking about a logarithmic spiral here and not something like an Archimedes spiral.

I don't know the difference.

The Archimedes spiral is what you get drawing with a string tied to a pencil and winding it around a central cylinder.  The logarithmic spiral has a constant angle at any point with a line drawn from that point to the center.  The latter is scale invariant. 

I

IMach0: The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames.
Mach1: Newton's gravitational constant G is a dynamical field.
Mach2: An isolated body in otherwise empty space has no inertia.
Mach3: Local inertial frames are affected by the cosmic motion and distribution of matter.
Mach4: The universe is spatially closed.
Mach5: The total energy, angular and linear momentum of the universe are zero.
Mach6: Inertial mass is affected by the global distribution of matter.
Mach7: If you take away all matter, there is no more space.
Mach8: 'Horseshoe, def above =, 4, pi symbol, funny p, G, T, squared' is a definite number, of order unity, where funny p is the mean density of matter in the universe, and T is the Hubble time.
Mach9: The theory contains no absolute elements.
Mach10: Overall rigid rotations and translations of a system are unobservable.

Mine do work:
Acceleration is relative.
Any group of objects treated collectively require a separate point of reference to define any form of motion.

Maybe I misunderstood the principle.  If Mach asserts that the laws of physics are the same in a rotating reference frame, then it is empirically wrong, but I don't see such assertions. This doesn't seem to be a description of our actual universe, but rather a mathematical treatment of the properties of abstract systems.  I'm talking about our actual universe in this thread, so all talk about there being only two elements in orbit say is irrelevant.  I said add reference stars if you need them to actually consider real physics. The rest of us don't.

An inertial frame is one without acceleration so you've written into your own definition that the age difference is caused by the fact that one accelerated and one didn't.

Two objects cannot separate and rejoin without one accelerating, so that in inherent in the situation at hand, not in my definition.
I'm just saying that all times can be computed purely as a function of the speeds of the participants in one arbitrary but consistent frame. I did not say that the computation must be done that way.

What? You only get that answer in the frame in which Earth is stationary. Both twins will have aged less in any other inertial frame, although the difference on their watches will of course be the same at the end in any frame, not just inertial ones.

The difference on their watches is all we care about, so we're good.

You were treating the Earth bound watch as stationary and the other watch as moving.

I was, but I don't have to.

I like to use acceleration alone to explain the difference.

That's fine. It's your preference. It's actually a reasonable way to explain the difference, whereas my goal was more to compute their ages at the end, not explain the discrepancy.

My method is simply to pick a frame (any frame) and stick with it.

Your method is incomplete and can't explain the difference in age without implying a preferred frame.

It gets the correct answers, so it is complete.  It doesn't require knowing the one actual correct frame or any absolute speeds of anything, so it doesn't imply a preferred one.

But that would be an absolute velocity relative to Bob.

You don't see the contradiction in this statement?

Yes I do, that's my point. Alice's velocity relative to Bob is purely coordinate dependant, because there's no change in separation over time.

Saying 'relative to Bob' identifies the coordinate system being used. In any other coordinate system, it wouldn't be 'relative to Bob', and yes, the difference in their velocities, and their separation, and a bunch of other stuff would be different.

You still seem to be in combat mode.

I'm trying to keep the topic on track, and it keeps being diverted to these abstract mathematical universes with different properties than ours.

Your last two posts are very disingenuous, deliberately taking what I say out of context in attempts to obscure the point. This is an example of what you're doing:

Dilation relative to a stationary observer, and discounting gravity, is entirely a function of speed and not acceleration.  The age of the twin in the twin experiment can be expressed as a function of integrating his speed, and acceleration has nothing to do with it.  Dilation relative to a non-inertial observer is a function of acceleration, but we're expressing Alice's orbiting clock relative to Bob here.

Did you mean dilation relative to a non-stationary observer?

...   no, I meant what I said.

But you said dilation relative to a stationary observer.

I said non-inertial observer, bolded above, and I meant non-inertial observer in that statement. You asked if I meant non-stationary, but one can be moving but still inertial, so I didn't mean that.  Non-inertial means accelerating.

So you meant dilation relative to an observer who is stationary relative to themselves?

I didn't follow most of the context around this bit, but I would typically not word things as you ask here, it being redundant. So I'd word it more as "dilation relative to an inertial reference frame".  It has nothing to do with observers or actually being observed. Everybody always puts named twins on the ships, but it only seems to serve to give them names, which can be done with a label maker.  Put a clock on board and that's enough.  It's real easy to get a clock up to .99c so long as you want only a few digits of precision.

Sigh. An accelerating observer takes into account the fact that they are time dilated and length contracted because of their acceleration. They can indirectly measure this by virtue of the fact that they can measure their own acceleration.

Not sure what you mean by this.  My accelerating ship has a proper length of 10 meters, and not sure what tools I have to show that it measures any other length.  I know it is accelerating via various clues like the accelerometer or the fact that water stays in my cup or that time at one end ticks faster than the other end.
OK, you're talking about something other than proper length, but I don't know how an accelerating observer might go about measuring anything except his proper length.
One can compute it: I am a meter tall in a frame where I'm moving fast, but there's no simple way to show that.

They then work out what the speed of light relative to themselves moving away in front of them would be if they were not time dilated and length contracted and they can see that the speed of light relative to themselves moving away in front of them slows as they increase their acceleration, with the reduction in speed of light relative to themselves moving away in front of them lessening with the same increase in acceleration as their overall acceleration increases in such a way that the speed of light relative to themselves moving away in front of them approaches without ever reaching 0.

I cannot parse this.  Maybe you could illustrate your method with an example. I'll pick one with nice high acceleration.
You, at time 0 shine a pulse of light outward, and immediately go chase it with a ship.  You accelerate for 45 days, 7 hours at 20g measured on ship clock.  After that much proper time, how far in front of the ship is the pulse of light in the final frame of the ship?
I computed it without any consideration of length contraction. I did the whole thing in the terminal frame of the ship, not in the initial frame. That seemed to simplify the calculations.  I got a bit over 2 light years away. Maybe I goofed, but that seems about right. You talk about light slowing in front of the ship, but I computed that it moved 2 light years in an eighth of a year, which is much faster than c.

An event horizon is a threshold in space beyond which no event can ever have a causal effect on you.  There is no such horizon in front of an accelerating object in the flat spacetime of the special relativity case.  Any event that happens in front of you (say a signal sent at you) will get to you, all the faster because you're accelerating towards it instead of just waiting for the light to make the trip.

I'm taking about a signal moving away from you.

OK, under SR conditions, there is no horizon in front of an accelerating craft beyond which an outgoing signal will never reach.  This is true even of a stationary thing, and yet light seems to move faster than c relative to an accelerating thing.

[A-wal (OP)]

If the two observers are moving away from each other then the observer who is spiralling can see the watch of the observer who isn't spiralling ticking at the same rate as their own watch. If the two observers are moving towards each other then the observer who isn't spiralling can see the watch of the observer who is spiralling ticking at the same rate as their own watch.

Agree.  We're talking about a logarithmic spiral here and not something like an Archimedes spiral.

I don't know the difference.

The Archimedes spiral is what you get drawing with a string tied to a pencil and winding it around a central cylinder.  The logarithmic spiral has a constant angle at any point with a line drawn from that point to the center.  The latter is scale invariant.

Okay, but why would an Archimedes spiral not work to reduce tick rate of the spiralling clock from the other observer's perspective?

IMach0: The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames.
Mach1: Newton's gravitational constant G is a dynamical field.
Mach2: An isolated body in otherwise empty space has no inertia.
Mach3: Local inertial frames are affected by the cosmic motion and distribution of matter.
Mach4: The universe is spatially closed.
Mach5: The total energy, angular and linear momentum of the universe are zero.
Mach6: Inertial mass is affected by the global distribution of matter.
Mach7: If you take away all matter, there is no more space.
Mach8: 'Horseshoe, def above =, 4, pi symbol, funny p, G, T, squared' is a definite number, of order unity, where funny p is the mean density of matter in the universe, and T is the Hubble time.
Mach9: The theory contains no absolute elements.
Mach10: Overall rigid rotations and translations of a system are unobservable.

Mine do work:
Acceleration is relative.
Any group of objects treated collectively require a separate point of reference to define any form of motion.

Maybe I misunderstood the principle.  If Mach asserts that the laws of physics are the same in a rotating reference frame, then it is empirically wrong, but I don't see such assertions.

That assertion is very much right at the heart of Mach's principle and is explicitly stated as Mach10: Overall rigid rotations and translations of a system are unobservable. Why would this be empirically wrong?

Rotation of an entire coordinate system is a contradiction, there's no rotation without considering a wider coordinate system and then it's not a rotation of the whole system. Mach9: The theory contains no absolute elements.

This doesn't seem to be a description of our actual universe, but rather a mathematical treatment of the properties of abstract systems.  I'm talking about our actual universe in this thread, so all talk about there being only two elements in orbit say is irrelevant.  I said add reference stars if you need them to actually consider real physics. The rest of us don't.

It is a description of our actual universe, and the abstract systems are used to demonstrate that although the acceleration and spin of objects can be objectively measured, this is only because of their accelerated and/or rotational motion relative to other objects in that system. It's beautiful.

Without it it's like, well motion is relative, but not really, not all of it, only purely inertial motion, which doesn't actually ever really exist. No, just no.

I'm trying to keep the topic on track, and it keeps being diverted to these abstract mathematical universes with different properties than ours.

Same properties, different perspective. You can make the whole universe the coordinate system and you're back to no possible motion including acceleration or spin.

Sigh. An accelerating observer takes into account the fact that they are time dilated and length contracted because of their acceleration. They can indirectly measure this by virtue of the fact that they can measure their own acceleration.

Not sure what you mean by this.  My accelerating ship has a proper length of 10 meters, and not sure what tools I have to show that it measures any other length.  I know it is accelerating via various clues like the accelerometer or the fact that water stays in my cup or that time at one end ticks faster than the other end.
OK, you're talking about something other than proper length, but I don't know how an accelerating observer might go about measuring anything except his proper length.
One can compute it: I am a meter tall in a frame where I'm moving fast, but there's no simple way to show that.

You could use the speeding up any inertial clock as a reference. You'd need to allow for the fact that its velocity relative to you is slowing the clock but you'd also need to account for Doppler shift and people assume that's done all the time in these kinds of scenarios.

They then work out what the speed of light relative to themselves moving away in front of them would be if they were not time dilated and length contracted and they can see that the speed of light relative to themselves moving away in front of them slows as they increase their acceleration, with the reduction in speed of light relative to themselves moving away in front of them lessening with the same increase in acceleration as their overall acceleration increases in such a way that the speed of light relative to themselves moving away in front of them approaches without ever reaching 0.

I cannot parse this.

I got frustrated and worded it very obtusely, sorry. I'll rewrite that paragraph.

They then work out what the speed of light relative to themselves moving away in front of them would be if they were not time dilated and length contracted and they can see it slows as they increase their acceleration, with the reduction of velocity lessening with the same increase in acceleration as their overall acceleration increases, in such a way that approaches without ever reaching 0.

This should describe a horizon in front of them that exactly mirrors the Rindler horizon behind them. Does the distance between an accelerating object and their Rindler horizon shortening by less distance with the same increase of acceleration as their acceleration increases match the velocity addition formula?

Maybe you could illustrate your method with an example. I'll pick one with nice high acceleration.
You, at time 0 shine a pulse of light outward, and immediately go chase it with a ship.  You accelerate for 45 days, 7 hours at 20g measured on ship clock.  After that much proper time, how far in front of the ship is the pulse of light in the final frame of the ship?
I computed it without any consideration of length contraction. I did the whole thing in the terminal frame of the ship, not in the initial frame. That seemed to simplify the calculations.  I got a bit over 2 light years away. Maybe I goofed, but that seems about right. You talk about light slowing in front of the ship, but I computed that it moved 2 light years in an eighth of a year, which is much faster than c.

You mean it traveled two light years in the inertial frame that the ship started in? That's calculating the extra velocity of the light based on the ship being time dilated and length contracted in this frame while they were accelerating.

I'm talking about the velocity of light as measured by them while they're accelerating not being greater than c despite their time dilation and length contraction.

I'm taking about a signal moving away from you.

OK, under SR conditions, there is no horizon in front of an accelerating craft beyond which an outgoing signal will never reach.  This is true even of a stationary thing, and yet light seems to move faster than c relative to an accelerating thing.

It only seems to move faster than c relative to an accelerating observer once they stop accelerating and use an inertial frame as a reference.

While they're accelerating, light maintains the same velocity relative to them despite the shortening of their length and slowing of their clock.

[Halc]

Okay, but why would an Archimedes spiral not work to reduce tick rate of the spiralling clock from the other observer's perspective?

The further out you go, the more the path resembles a perfect circle: All dilation, no Doppler.  Close to the center, it becomes the opposite.

If Mach asserts that the laws of physics are the same in a rotating reference frame...

That assertion is very much right at the heart of Mach's principle and is explicitly stated as Mach10: Overall rigid rotations and translations of a system are unobservable. Why would this be empirically wrong?

Because I can observe if I am rotating, even if enclosed in a box that rotates with me. I notice you qualify rotation with the the word 'rigid', but there are ways to observe rotation even confined to rigid objects, such as is employed by a ring interferometer.

They then work out what the speed of light relative to themselves moving away in front of them would be if they were not time dilated and length contracted

How might I go about that?  I would need to choose some arbitrary frame that probably has nothing to do with my present state. Length contraction is a relation, not a property, yet you are treating it as a property in this statement, hence it makes no sense.

Maybe you could illustrate your method with an example. I'll pick one with nice high acceleration.
You, at time 0 shine a pulse of light outward, and immediately go chase it with a ship.  You accelerate for 45 days, 7 hours at 20g measured on ship clock.  After that much proper time, how far in front of the ship is the pulse of light in the final frame of the ship?
I computed it without any consideration of length contraction. I did the whole thing in the terminal frame of the ship, not in the initial frame. That seemed to simplify the calculations.  I got a bit over 2 light years away. Maybe I goofed, but that seems about right. You talk about light slowing in front of the ship, but I computed that it moved 2 light years in an eighth of a year, which is much faster than c.

You mean it traveled two light years in the inertial frame that the ship started in?

The ship might have been accelerating for an indefinite time, so there is no such frame. I've been sending out pulses ten times a day (proper time), and that's just one of them.  That one is 2 light years away in the current frame of when the ship clock reads 45.3 days.  Maybe it would be easier for you to think of the ship clock currently reading zero and that particular pulse was sent 45.3 proper days prior.

I'm talking about the velocity of light as measured by them while they're accelerating

What do those words mean?  If not velocity of light relative to the IRF in which the accelerating object is currently stationary (which is always just c), then it needs to be the difference of where the light is now and where it was a moment ago.  That distance is of course frame dependent, but if you're not using the frame of the object from moment to moment, then it isn't really a speed relative to the accelerating object.
I didn't compute the where it was a moment ago.  I computed where it is now (>2 LY away) vs where it was 45 days ago (at the accelerating object).  It's not going a constant speed, so that's its average speed (> 16c) over a month and a half. It is always increasing, which is why I gave it some time to get decently away.

It only seems to move faster than c relative to an accelerating observer once they stop accelerating and use an inertial frame as a reference.

And what frame do we use if I don't stop, since where it is now has nothing to do with what I plan to do tomorrow?

[A-wal (OP)]

Okay, but why would an Archimedes spiral not work to reduce tick rate of the spiralling clock from the other observer's perspective?

The further out you go, the more the path resembles a perfect circle: All dilation, no Doppler.  Close to the center, it becomes the opposite.

So it's not that you couldn't use that kind of spiral, it's just that it's awkward and pointless in this situation.

If Mach asserts that the laws of physics are the same in a rotating reference frame...

That assertion is very much right at the heart of Mach's principle and is explicitly stated as Mach10: Overall rigid rotations and translations of a system are unobservable. Why would this be empirically wrong?

Because I can observe if I am rotating, even if enclosed in a box that rotates with me. I notice you qualify rotation with the the word 'rigid', but there are ways to observe rotation even confined to rigid objects, such as is employed by a ring interferometer.

That's not a rigid rotation, rigid rotation is the whole coordinate system. That's a measurement of rotation relative to more distant objects, thereby enlarging the coordinate system. An interferometer would not be able to measure a rigid rotation of the universe as a whole.

They then work out what the speed of light relative to themselves moving away in front of them would be if they were not time dilated and length contracted

How might I go about that?  I would need to choose some arbitrary frame that probably has nothing to do with my present state. Length contraction is a relation, not a property, yet you are treating it as a property in this statement, hence it makes no sense.

Start off inertial and use a clock in that frame as a reference and as you accelerate you'd need to compensate for the inertial clock's time dilation caused by your velocity relative to it to work out how slowed your own clock is by how sped up the inertial clock is. You'd have to account for Doppler shift as well, but like I said that's assumed most of the time anyway.

None of this really matters, you can simple measure g-force to get an objective measurement of your acceleration.

Maybe you could illustrate your method with an example. I'll pick one with nice high acceleration.
You, at time 0 shine a pulse of light outward, and immediately go chase it with a ship.  You accelerate for 45 days, 7 hours at 20g measured on ship clock.  After that much proper time, how far in front of the ship is the pulse of light in the final frame of the ship?
I computed it without any consideration of length contraction. I did the whole thing in the terminal frame of the ship, not in the initial frame. That seemed to simplify the calculations.  I got a bit over 2 light years away. Maybe I goofed, but that seems about right. You talk about light slowing in front of the ship, but I computed that it moved 2 light years in an eighth of a year, which is much faster than c.

You mean it traveled two light years in the inertial frame that the ship started in?

The ship might have been accelerating for an indefinite time, so there is no such frame. I've been sending out pulses ten times a day (proper time), and that's just one of them.  That one is 2 light years away in the current frame of when the ship clock reads 45.3 days.  Maybe it would be easier for you to think of the ship clock currently reading zero and that particular pulse was sent 45.3 proper days prior.

I'm talking about the velocity of light as measured by them while they're accelerating

What do those words mean?  If not velocity of light relative to the IRF in which the accelerating object is currently stationary (which is always just c), then it needs to be the difference of where the light is now and where it was a moment ago.  That distance is of course frame dependent, but if you're not using the frame of the object from moment to moment, then it isn't really a speed relative to the accelerating object.
I didn't compute the where it was a moment ago.  I computed where it is now (>2 LY away) vs where it was 45 days ago (at the accelerating object).  It's not going a constant speed, so that's its average speed (> 16c) over a month and a half. It is always increasing, which is why I gave it some time to get decently away.

It only seems to move faster than c relative to an accelerating observer once they stop accelerating and use an inertial frame as a reference.

And what frame do we use if I don't stop, since where it is now has nothing to do with what I plan to do tomorrow?

It's actually very simple.

If you start off in an inertial frame and then accelerate to half the speed of light relative to an object at rest in your starting frame then you still measure the speed of light as c in your new inertial frame because you are length contracted and time dilated from the perspective of the object at rest in your starting frame.

If you use an accelerometer you can measure the slowing of the speed of light as you accelerate due to its speed relative to you remaining constant despite your length contraction and time dilation caused by your acceleration.

If the speed of light is measured in this way then its reduction of velocity should lessen with the same increase in acceleration as your overall acceleration increases, in such a way that approaches without ever reaching 0.

This should follow the velocity addition formula. It mirrors velocity relative to massive objects but with acceleration relative to the velocity of light, so this regard acceleration can be viewed as velocity relative to energy.

This should mirror how the distance between an accelerating object and its Rindler horizon shortens by less distance with the same increase of acceleration as the object's acceleration increases.

[Janus]

If you start off in an inertial frame and then accelerate to half the speed of light relative to an object at rest in your starting frame then you still measure the speed of light as c in your new inertial frame because you are length contracted and time dilated from the perspective of the object at rest in your starting frame.

No.  What another frame measures in terms of length contraction or time dilation for you, has no bearing what measurements you make. 

 According to Frame A, a light ray is sent from from one clock to another clock (both at rest relative to Frame A) 1 light min away and synchronized to the first clock . The light leaves the first clock when both clocks read 0 and arrives at the second clock when both clocks read 1 min.
You are flying by at 0.8 c relative to Frame A opposite to the direction of the light ray.
Frame A would say that you are time dilated and length contracted.
But as far as you are concerned, that is neither here nor there.
You measure that is the two that clocks that are moving at 0.8c (same direction as the light), are time dilated (running 0.6 as fast as your own), and you will measure that the clocks and the distance between them is length contracted ( the distance between the clocks is 0.6 light min.)
You will also note that the receiving clock is 0.8 min behind the sending clock.
None of this is due to "your" motion, but due to their motion relative to you.
You will however agree that the light leaves the sending clock when it reads 0 and arrives at the receiving clock when it reads 1 min.
Thus according to you, when the light leaves the sending clock, the receiving clock reads -0.8 min and advances to 1 min in the time that it takes the light to pass from one clock to another.  In other words, it ticks off  1.8 min.  Since this clock ticks 0.6 as fast as your own, this means that 3 min passes on your clock. 
In that 3 min, the pair of clocks will have moved 3 min x 0.8c = 2.4 light min.  Since the receiving clock had a 0.6 light min "head start", the total distance traveled by the light  relative to you, as measured by you is  2.4 + 0.6 = 3 light min, which it covered in 3 min, meaning it traveled at c.
All worked out without once considering what Frame A said was happening to your clock or length.

[A-wal (OP)]

If you start off in an inertial frame and then accelerate to half the speed of light relative to an object at rest in your starting frame then you still measure the speed of light as c in your new inertial frame because you are length contracted and time dilated from the perspective of the object at rest in your starting frame.

No.  What another frame measures in terms of length contraction or time dilation for you, has no bearing what measurements you make.

Er, yes. You've grabbed one bit of what I said, taken it completely out of context and then gone on to explain something I already know full well.

None of this is due to "your" motion, but due to their motion relative to you.

No. What another frame measures as your velocity relative to them, you will measure as their velocity relative to you.

Neither observer is exclusively "in motion", from the other observer's perspective it is you who is in motion. Inertial motion is purely frame dependent, so they are in motion relative to each other.
An object is moving at away from you at 0.5c relative to your frame.
Now if we switch to their frame you are moving at 0.5c away from them.
But that in no way contradicts what your own frame.
You measure that the other object is moving at 0.5c relative to you (the same speed that you are moving from their perspective) and they measure that you are moving at 0.5c relative to them (the same speed that they are moving from your perspective). The relative velocity between you is 0.5c.
So you can see that inertial motion is not unique to either observer, but relative.
None of this is due to "your" motion, but due to the relative motion between you.
If an observer has a velocity relative to you then you will always agree on that velocity because it's the same for both of you.
So if they were to accelerate for time in the same direction as the relative motion from your perspective so that they're now steadily moving away from you at say 0.8c, then from their perspective you are moving away from them at 0.8c despite the fact that they are the ones who accelerated from both perspectives.
During that acceleration the velocity is increasing for both observers . From your perspective their velocity moving away from you is increasing due to their acceleration, but they see your velocity moving away from them increasing, because it's the velocity that you're moving away from each other.
All worked out without once considering which one was actually the one moving and which was at rest.


Let's try this again.

If you start off in an inertial frame and then accelerate to half the speed of light relative to an object at rest in your starting frame then you still measure the speed of light as c in your new inertial frame because you are length contracted and time dilated from the perspective of the object at rest in your starting frame.

If an object is moving away from you at 0.8c c and you accelerate towards it then its relative velocity away from you decreases. If you stop accelerating so that it's now moving away from you at 0.4c then that same amount acceleration will not decrease the relative velocity to zero because of the velocity addition formula.

Velocity increases relative to other objects lessen with the same acceleration as your velocity relative to them increases, in such a way that your relative velocity approaches without ever reaching c. This is how velocity relative to mass works, because of length contraction and time dilation.

Same again, an object is moving away from you at 0.8c c and you accelerate towards it and its relative velocity away from you decreases. If you stop accelerating so that it's now moving away from you at 0.4c but light is still moving at the same velocity relative to you as it was when they were moving away at 0.8c.

This is also caused by length contraction and time dilation, it's what keeps the speed of light constant. Now when you are accelerating you are time dilated and length contracted from the perspective of an inertial observer. Despite that you still measure the speed of light moving away in from of you as c.

This can be viewed as a slowing of light relative to you due to your acceleration in the same direction. Measured in this way, its reduction of velocity should lessen with the same increase in acceleration as your overall acceleration increases, in such a way that approaches without ever reaching 0.

This is how velocity relative to energy works. With the same velocity addition formula describing an accelerating object's velocity increasing by lesser amount with the same increase in acceleration (rather than velocity) in such a way that their velocity approaches without ever reaching c.

[Halc]

Let's try this again.

Everything below you saying this is 1) off topic from the title question and 2) obviously showing no intention of actually paying attention to the responses being received.  So there seems to be little point in replying to most of it.

Now when you are accelerating you are time dilated and length contracted from the perspective of an inertial observer.

Then it is the state of things according to this hypothetical inertial observer, not according to you. That seems to be the point your're not getting.  Sure, I can make me and some distant thing as close as I want by considering it from the perspective of some 3rd irrelevant frame, except there is no event horizon in that inertial frame. I understand what you're saying in your post, but you're just looking at something from a perspective other than the current or accelerating frame of the object in question.

This is how velocity relative to energy works.

Velocity relative to energy?  Really?

[A-wal (OP)]

Let's try this again.

Everything below you saying this is 1) off topic from the title question and 2) obviously showing no intention of actually paying attention to the responses being received.  So there seems to be little point in replying to most of it.

Seriously? I'm paying attention to every reply. Most of what I say on the other hand seems to be taken completely out of context.

Now when you are accelerating you are time dilated and length contracted from the perspective of an inertial observer.

Then it is the state of things according to this hypothetical inertial observer, not according to you. That seems to be the point you're not getting.

I totally get that. I'm using an inertial observer's perspective, so how could I possibly not understand that it's from that inertial observer's perspective?

Since when is it not okay to compare different reference frames in relativity?

Sure, I can make me and some distant thing as close as I want by considering it from the perspective of some 3rd irrelevant frame, except there is no event horizon in that inertial frame. I understand what you're saying in your post, but you're just looking at something from a perspective other than the current or accelerating frame of the object in question.

Yes I know exactly what I'm doing, and it doesn't in any way detract from the point I'm making.

This is how velocity relative to energy works.

Velocity relative to energy?  Really?

Yes really. That is one way of looking at it.

An object's velocity relative to matter changes when they accelerate while the velocity of energy (not just light) stays the same in all inertial frames. This is because time dilation and length contraction keep the speed of light the same while velocity relative to massive objects varies from inertial frame to inertial frame.

Acceleration creates a time dilation and length contraction doesn't affect the speed of light relative to the accelerating observer. In that sense the light moving away in front of you is slowed due to your acceleration in the same direction and so your acceleration can be thought of as your velocity relative to that light.

If you agree with how I'm wording it or not, as long as you understand what I mean then you should understand my point that (when viewed in this way) the reduction of the speed of light relative to the accelerating observer lessens due to the same increase of acceleration as their overall acceleration increases.

In this way it matches the velocity addition formula and the distance between the accelerating observer and their Rindler horizon should be reduced in exactly the same way, with the distance decreasing by a lesser amount with the same increase in acceleration as their overall acceleration increases.