[Mike Gale (OP)]

Vacuum permittivity (eo) and permeability (uo) arise in the context of the Coulomb force. They represent the elasticity of spacetime and are completely defined in terms of light speed and pi:
uo=4*pi/10^6
eo=1/c^2/uo
It's analogous to waves on a guitar string, for which the speed of propagation depends on string tension. The seemingly arbitrary definition of uo is an artifact of the units of measure.

[Mike Gale (OP)]

I removed the angular coordinates from the metric in my last edit to Reply#57 because I realized it's not that simple for orbital free fall. The coordinate speed of light is more complicated.in that case because the velocity vector for the SR correction changes direction over time.

[timey]

Yes - that is so!  ... it will change over time and distance.

[Mike Gale (OP)]

I think we can recover simplicity by switching back to Cartesian coordinates:
(cdt')² - dx'² - dy'² -dz'² = (c'dt)² - dx² - dy² -dz²
The complexity is then in the coordinate transforms because the speed of light depends on the direction of motion.

[timey]

So's not to clog up Jeff's thread as we are getting off topic there...

Well since relativistic effects have been experimentally confirmed at speeds under 30 miles an hour, then I think we can ditch the classical approach
And if it is momentum that changes with v = a, then in the case of free fall surely the energy change in momentum is due to that which is the cause of the acceleration, and not that which is being accelerated.

It's a bit of both. Mass increases with velocity and velocity increases with acceleration. That's why GR theorists prefer 4-momentum. It reminds them that relativistic mass is a consequence of spacetime dilation. It's typical insider terminology though. They want to make you feel stupid if you prefer to think in terms of relativistic mass. It's so passe.

Space time dilation of space, or space time dilation of time?

Both. That's what SR teaches us.

So when you employ coordinate speeds for light that's GR time dilation related?

[timey]

So when you employ coordinate speeds for light is this GR time dilation related?

[Mike Gale (OP)]

Yes. SR dilation is due to velocity. GR dilation is traditionally thought of as a mixture of velocity and acceleration, but I contend that it should be exclusively acceleration.

[timey]

So - if your coordinate speed of light is due to GR time dilation, then the percentage of the speed of light that m's velocity is, as per held relative to a standard second, will be decreasing.
Therefore the SR time dilation that is caused by your velocity will also be decreased, as per that reference frame.

But where the fu*k are you?

SR length contraction and the reciprocal dilation of space will render your coordinates, and therefore the relevant coordinate speed of light difficult to define!

Correct?

[timey]

I'm going to reformulate what I've said last post a little more coherently:

If your coordinate speed of light is due to GR time dilation - then in the reference frames where the speed of light is held relative to the shorter seconds at h from M = Earth (longer seconds for inbound to BH), m's constant velocity as per held relative to a standard second* will be a decreasing (increasing for BH) percentage of the reference frame that m is passing through's speed of light, and this will affect the SR measurements.

*Unless you state m's speed relative to the shorter second (longer second for BH) of the reference frame, in which case m's speed as per a standard second will be increasing (decreasing for BH), also affecting the SR measurements.

SR holds all of its time and length measurements relative to the speed of light per standard second - but by the remit of a speed of light held relative to the shorter seconds (longer seconds for BH) of the reference frames m is passing through, the SR time dilation that m will be experiencing, (or appears to be experiencing, dependent on your school of thought*) is decreasing (increasing for BH) as the speed of light of each reference frame is held relative to shorter seconds (longer seconds for BH)
(This being true unless you have decided that m's speed is increasing (decreasing for BH) as per the standard second when passing through reference frames with shorter seconds (longer seconds for BH))

*In which case, as per the notion that the shorter seconds (longer seconds for BH) of GR time dilation are affecting m's speed, is the speed m is travelling at also affected by SR time dilation?

But whichever you decide, where the fu*k are you?

SR length contraction and the reciprocal dilation of space will render your coordinates, and therefore the relevant coordinate speed of light difficult to define!

Correct?

[timey]

Not being able to 'know' exactly where one is in space is what led Katherine G Johnson to utilising Euler's theorem (?) in order to land space shuttle within 20 mile radius, and subsequently launch John Glenn into orbit?

[Mike Gale (OP)]

Locating oneself in a relativistic context is admittedly a tricky business. Bell's spaceship paradox is a prime example. However, the paradox arises from philosophy, not physics. The equations are perfectly self consistent. It is only the interpretation of the result that is contentious. To be more specific, questions of where turn into questions of where and when. Simultaneity is in the eye of the beholder.

[Mike Gale (OP)]

Not being able to 'know' exactly where one is in space is what led Katherine G Johnson to utilising Euler's theorem (?) in order to land space shuttle within 20 mile radius, and subsequently launch John Glenn into orbit?

Probably Kepler. Euler was into fluid dynamics.

[timey]

Nope - 'twas Euler's method for definite. I don't make a statement without researching it.  Describes curves I believe...

[jeffreyH]

https://en.m.wikipedia.org/wiki/Euler_method

[Mike Gale (OP)]

Ah yes. Numerical integration technique. Math, not physics. That came up in the NASA movie "Hidden Figures." It's considered "old school" because computers do that for us nowadays and they can often find exact solutions using symbolic integration techniques.

[timey]

Yes that's right.  When I first made the post I pulled the term Euler's theorem out of my memory of watching the movie 'Hidden Figures', which is why the word theorem had (?) after it.

Sorry, I realise that saying theorem was a bit misleading.  After getting interested by the memory and further investigating, I corrected myself on my own 'my model of a cyclic universe continued again' thread, where I also made mention, but forgot to do so on your thread.

So yes, Katherine Johnson had to resort to using Euler's method when calculating the trajectory of the space mission in order to understand the exact position in space that the shuttle would be located - that it would be then possible to calculate a re-entry and specific landing zone.

[Mike Gale (OP)]

The point being that Johnson's calculation had nothing to do with SR or GR. It was all Newton and the fact that she had to resort to numerical integration techniques is telling both of the power of Newtonian dynamics and of the complexity of the problem. It is a common misconception that classical dynamics is easy compared to relativity. It's not. Relativity may be harder to visualize, but the driving factor for complexity is the number of objects involved in the calculation.

[jeffreyH]

An example of this is the three body or N body problems.

[Mike Gale (OP)]

That's right. N-body problems are notoriously difficult to solve (for N>2) even before you invoke relativity. I should correct myself on one point though. It is the power of Newtonian dynamics (as opposed to its complexity) that is exemplified by the fact that Johnson didn't need relativity to get John Glenn home.

[timey]

Relativity was already a theory when Johnson made her calculation, and time shift was already a phenomenon before Einstein formulated Relativity.
There is no way that any trajectory that Johnson calculated would or could have not taken this time shift into account.

The point is that while it was possible to know how to calculate going up, and to know how to calculate an orbit, it was necessary to make a calculation of a combination of both in order to launch into orbit with a specific re-entry zone in mind.

This is where Johnson employed Euler's method and invented new mathematics in order to calculate trajectories for the initial NASA space missions.

But to say so - I've completely lost my train of thought as to why it may of been useful for your relativistic correction thread.  I know why I'm interested, and if I remember (might have to read back on the thread for that), for what reason I thought it may have been relevant to you, I'll be back.