[timey]

But when not in free fall?

Do people in uniform motion in vastly differing gravity potentials agree on the speed of light, where clearly they will be unable to hold the speed of light relative to a standard second, but will have to hold the speed of light relative to the length of second of their gravity potential location?

[Mike Gale (OP)]

They do not agree. The speed of light on the surface of the moon is different than the speed of light on the surface of the Earth for example, even if you run along the surface of the Earth to keep pace with the moon's orbit. GPS is a prime example of that, although there is a lot of confusion about the magnitude of the effect in the literature.

[timey]

Good good.

So - how is it possible to move light across space with SR, when SR is holding the speed of light relative to a standard second?

[Mike Gale (OP)]

Your "standard second" is presumably defined in a reference frame that is stationary with respect to the gravitating mass when the observer is infinitely far away. That's a fair description of the space between galaxies if you ignore the Hubble effect. In that case, the speed of light is c. Light is going as fast as it possibly can. Barring the exterior of white holes and the interior of black holes, there is no reference frame in which it can go any faster.

[timey]

I don't like 'presumably' or 'infinity far away' much... but oh well for the mo.

Going back to moving mass across differing gravity potentials with SR.

When SR makes its length contraction calculations we are talking about 0.806c (might have got that figure wrong but can't be arsed to look it up) causing a 50% length contraction.

But if the length of second in a reference frame is differing from a standard second, then the percentage of the speed of light will not be 0.806 and the length contraction will not be 50%!

[Mike Gale (OP)]

If you're talking about the NIST second, it is defined in terms of the frequency of oscillation of a cesium 133 atom on Earth (at sea level or in free fall. I can't remember which.) The vacuum speed of light, which is that measured by an infinitely removed observer or any other observer in free fall, then defines the standard meter. The cesium frequency is an arbitrary choice, but it's a good one because it's very stable in any given reference frame.

[timey]

According to SI units the metre is defined as per the speed of light relative to the SI unit of the standard second, as per the oscillation of a caesium atom.

The distance of a metre 'is' now actually defined by the speed of light.

It is true that a caesium atomic clock can be calibrated to oscillate to the tune of a standard second at any gravity potential, but once calibrated, a cession atomic clock will oscillate at a higher frequency placed in a higher gravity potential, and at a lower frequency when placed in a lower gravity potential.

So - when moving mass across differing gravity potentials, the speed of light will, if the equivalence principle is upheld be differing because of the differing length of second, and the percentage of the speed of light that the mass is moving at in reference frames of differing lengths of second will be changing, and the length contraction will be altered, won't it?

[timey]

It is of interest that the caesium atomic clock has only been tested under lab conditions*, and that those lab conditions have been restricted as to the gravity potential of the co-ordinates of the lab itself...

(*Edit: Although caesium atomic clocks are (I think) used on satellites with respect to GPS, the actual rigorous testing of gravity potential considerations have been conducted under lab conditions.)

Quantum clocks are far more sensitive than the caesium atomic clock, but are so far unreliable and have to be used in conjunction with the caesium atomic clock in order to be of any use.

The caesium atomic clock itself 'can' be more sensitive - but the physical dimensions of the clock, in that the atoms are being processed in a 3 foot high chamber, negate any further sensitivity because the atoms will change frequency within their movement in the dimensions of that chamber.  Therefore the caesium atomic clock is 'dumbed down' by the choice of microwave frequency used to excite the caesium atoms by passing them through the microwave beam on their upward trajectory.

NIST do have plans for a portable caesium atomic clock, which I am waiting for with bated breath.  I am almost sure that it will be possible to standardise a second at sea level earth and achieve an absolute that can be used to relate the difference any different gravity potential may have against.

I also am very interested in whether or not a geological difference in mass density will affect the clock if 2 clocks were to be placed in the same gravity potential, but at locations of vastly differing geological density...
It is my thoughts that gravity potential may affect a clock in one way, but that the difference in mass density will affect it oppositely... where the clock in the denser location will run faster.

Going back to SR considerations being altered by an invocation of GR in relation to the equivalence principle...  Do you have any thoughts?

Where I have run through the scenario myself, (not mathematically, but geometrically), I encounter dimensional problems...

[Mike Gale (OP)]

Wait a tick. The frequency at which a cesium atom oscillates depends only on the local field strength (electromagnetic and gravitational) and temperature. It's not like a quartz clock, where the frequency of oscillation is controlled by applying pressure to the crystal. It is an oscillation between two ground states, which have equal probability in a benign environment. Each transition releases a photon, which can be detected. The elapsed time between detection events defines the second, not the other way around. From the perspective of a distant observer, frequency decreases with increasing field strength so a local second as perceived from afar is longer if the clock is experiencing gravity. This is evident from the metric because dT=dt/sqrt(1-rs/r) if dr=r*d(angle)=0. GPS compares clocks in free fall with clocks on the ground so there can be no doubt about the validity of this model. There have been several experiments with clocks at elevation, too. In fact, standard time is routinely corrected for variations in elevation due to seismic activity.
I don't know what you mean by "quantum clock", but that would be a fair assessment of a cesium clock. That is, they are one and the same. Again though, I don't understand your obsession with the "standard" second. In the absence of gravity, there is no preferred reference frame. The choice is entirely arbitrary and has no impact on the physics.

[Mike Gale (OP)]

And again, I must insist that we focus this discussion on the topic at hand. If you want to debate concepts like the standard second, you (timey) should open another thread. Send me a PM if you want my input. I'm happy to oblige. I find these kinds of discussions instructive (to myself) because it forces me to formulate my thoughts in non-technical terms. I think it was Einstein who said something like, "if you can't explain it in everyday terms, you don't understand it yourself."

[Mike Gale (OP)]

To that end, there are (I think) 3 topics on the table:
1) Implications of the new metric (e.g. for Big Bang and Hubble.)
2) Implications of complex spacetime (e.g. for QM.)
3) The form of the coordinate transform for a GR metric.

[Mike Gale (OP)]

I think Flamm's paraboloid is the right way to think about the coordinate transform:
t'=t*sqrt(1-rs/r)
r'=r/sqrt(1-rs/r)
angle'=angle
Does that make sense? I keep getting confused between the local free fall frame and the local stationary frame. I think these transforms apply to the latter. Unlike the Lorentz transform, there is no mixing of time and space (because the reference frames are stationary with respect to one another.) The equivalent for the new metric is:
t'=+/-t*(1-rs/2r)
r'=+/-r/(1-rs/2r)
angle'=angle
Where the sign is positive for r>rs and negative otherwise. I guess that would preclude complex spacetime. Bad news for Penrose.

[Mike Gale (OP)]

I think the white hole scenario corresponds to the negative of the scaling distance. Does that make sense? Any Big Bang aficionados out there?

[timey]

Wait a tick!

Very slick!

And again, I must insist that we focus this discussion on the topic at hand. If you want to debate concepts like the standard second, you (timey) should open another thread. Send me a PM if you want my input. I'm happy to oblige. I find these kinds of discussions instructive (to myself) because it forces me to formulate my thoughts in non-technical terms. I think it was Einstein who said something like, "if you can't explain it in everyday terms, you don't understand it yourself."

The choice of which question you choose to answer might be having a bearing on the 'focus' me thinks.

For instance in relation to this:

Back to business though. Does it strike anyone as odd that there is no equivalent to the Lorentz transform for the SC metric?

You might have chosen to answer this:

Going back to moving mass across differing gravity potentials with SR.

When SR makes its length contraction calculations we are talking about 0.806c (might have got that figure wrong but can't be arsed to look it up) causing a 50% length contraction.

But if the length of second in a reference frame is differing from a standard second, then the percentage of the speed of light (edit: that the mass is travelling at) will not be 0.806c in that reference frame, and the length contraction will not be 50%!

And with respect to this:

I don't understand your obsession with the "standard" second. In the absence of gravity, there is no preferred reference frame. The choice is entirely arbitrary and has no impact on the physics.

You might consider this:

I am certain that our level of understanding has a lot of room for growth. There are a lot of known unknowns, but almost surely more unknown unknowns.

My enjoyment of physics is in exploring the known unknowns, and the unknown unknowns.

My fascination with the standard second:

https://www.thenakedscientists.com/forum/index.php?topic=69800.0

[Mike Gale (OP)]

I think my coordinate transform is wrong. It has to satisfy this line element:
(cdt')^2 - (dr')^2 = (c'dt)^2 - (dr)^2
The primed coordinates are those in the Lorentz transform and c' is the coordinate speed of light in the SC solution:
cdt'=(c'dt-vdr/c')*gamma
dr'=(dr-vdt)*gamma
c'=c*(1-rs/r) for old metric or c'=c*(1-rs/2r)^2 for the new one.
Note that gamma is the same in both reference frames because v/c'=v'/c.
The equation reduces to the Minkowski metric when rs=0 but in order for it to reduce to the SC metric when v=0, gamma has to morph into 1/sqrt(1-rs/r) so that:
cdt'=c'dt/sqrt(1-rs/r)=cdt*sqrt(1-rs/r)
dr'=dr/sqrt(1-rs/r)
That seems to do the trick, but plugging these values into the line element gives the coordinate speed of light. I did not expect that to happen, but I guess that makes sense. What it's saying is that there is no mixing of time and space in the suspended reference frame. Spacetime is simply squashed in the radial direction. The mixing (i.e. rotation or pinch) only happens when you get moving.
In practical terms, it means that gamma must be interpreted in terms of velocity for the free fall case and scaling distance for the suspended case. Anything in between is a mixture of the two and that's when the local observer feels the force of acceleration. In other words, force is due to the difference between (1-rs/r) and (1-v^2/c^2). The next step then is to express that relationship in mathematical terms.

[Mike Gale (OP)]

Well, that wasn't as hard as I imagined now that I think about. These scaling factors are, after all, just expressions of energy: potential energy in the case of GR and kinetic energy in the case of SR. The old metric ambiguates its roots because it is formulated in a classical context. Multiplying the old scaling factor by mc^2 produces a unrecognizable result:
mc^2 - 2GMm/r
Multiplying the new scaling factor by m^2c^4 gives something more familiar:
mc^2 - GMm/r = mo*c^2
This is the expression for relativistic mass in the free fall case, which is where this discussion thread started. The force one feels when accelerating is due to changes in relativistic mass, which itself is due to the spatial gradient of the potential.

[Mike Gale (OP)]

So we have come full circle. The old metric leads to nonsense when exposed to SR. The new one is perfectly consistent with SR. Can there be any doubt?

[Mike Gale (OP)]

What we have now is yet another new metric:
(cdt')^2 - (dr')^2 = (c'dt)^2 - (dr)^2
where:
cdt'=(c'dt-vdr/c')*gamma
dr'=(dr-vdt)*gamma
gamma=1/sqrt(1-v^2/c'^2)=1/sqrt(1-v'^2/c^2)
c'=c*(1-rs/2r)^2~c*(1-rs/r)
It reduces to SR (i.e. uniform motion) when rs=0 and GR (viz. free fall) when rs/r=v^2/c'^2. The equality does not hold in any other case because, if you're fighting the force of gravity with a rocket engine for example, you are adding energy to the equation. Nobel prizes all around if anyone can come up with a testable consequence of this. Or they might bend that rule if this metric can be related to QM.
See also Re: Relativistic Correction to the Schwarzschild Metric

[Mike Gale (OP)]

With regards to QM, the line element must be related to the argument of the exponent in the De Broglie wave equation:
Psi=exp(i*mv/hbar*(x-vt))
The idea would be that the line element represents the magnitude of a complex vector:
dz=dr+ic'dt
dz'=dr'+icdt'=(dr-vdt)*gamma+i(c'dt-vdr/c')*gamma

[timey]

Crawling on all fours round your maths as a toddler who has not yet learned to walk, (or talk), could I please just ask you this Mike:

You see where you say about this hbar being related to vacuum permittivity - is vacuum permittivity relating to 'spatially' variable spaces in space?


Edit:  Where Mike has now edited his post, it will be extremely difficult to see how my question relates!