Moving faster but taking longer

[DrGreg]

As a curiousity, if the flat-earthists were right, i.e. if the Earth were not a sphere but an infinite flat plane, I think I'm right to say that the equivalence principle would be exactly true and not just locally true. Someone freely falling towards an infinite flat Earth would have no way of detecting gravity at all, there would be no tidal effects and technically the spacetime curvature would be zero!

 

[gonegahgah]

That sounds okay to me. How does it sound to everyone else?

Again - based on the dispersing wave analogy - I don't think gravity would decrease at all for any distance from the infinite flat surface giving - as Dr Greg says - zero space time curvature. ie. All things would fall in a straight co-ordinate line; rather than approaching quicker and quicker like ours does.

That is because the gravity wave would be generated from all points of the infinite flat plane and spreads onwards in tandem. Even if it were to sub-spread as an expanding part sphere arc square the surrounding sub-spread would reinforce it back up to full strength.

How curious? All imaginary of course.

 

[gonegahgah]

Actually you would still fall with a curve. The force acting on you for the flat Earth is always proportional to 1/r⁰ but it acts none-the-less.

 

[DrGreg]

Actually you would still fall with a curve. The force acting on you for the flat Earth is always proportional to 1/r⁰ but it acts none-the-less.

Certainly free-falling objects accelerate towards an infinite flat Earth (as measured by someone on Earth). But they accelerate in such a way that objects that are initially stationary relative to each other will remain stationary -- i.e. no tidal forces. (And in general, all free-falling objects move at constant velocities relative to each other. That's pretty much the definition of flat spacetime.)

Note, however, from the point of view of someone on Earth, the free-falling objects are not a fixed distance apart, due to continuously-changing length contraction, which means that the acceleration of each object relative to Earth differs. In other words, the "acceleration due to gravity" still varies with height despite the fact that spacetime is flat.

 

[Ich]

That thing could be described by mirrored Rindler coordinates?

 

[DrGreg]

That thing could be described by mirrored Rindler coordinates?

Yes.

However, when I claimed that an infinite flat slab would result in flat spacetime I was relying on my memory. A little googling has left me confused.

Post #14 of this thread in this forum seems to suggest I might be wrong, but flat spacetime might instead be produced inside an off-centre spherical hole inside a uniform spherical planet!

On the other hand, I think equation (1) of this page of mathpages.com indicates I was right after all?

Everything I said in previous posts is true in the relativistic version of a "uniform gravitational field". What is no longer clear to me is whether there is a hypothetical distribution of matter that would be capable of generating such a field.

 

[JesseM]

I see what you mean.

The other insurmountable challenge I see is creating an equivalent observer & satellite scenario either; if we wanted to try.
Say we replace the person and box with a GPS and place an observer on the 'ground' in the examples.
Unfortunately we can not create an equivalent scenario between gravity and acceleration for this because we will not be able to create a constant equivalent for the observer on the 'ground' between the two types (ie acceleration vs gravity).

The other problem of course is that it is probably impossible to simulate an orbit scenario using acceleration anyway? That is probably correct; is it?

So it would be difficult (impossible?) to model the GPS using acceleration instead of gravity? So unfortunately we couldn't compare them this way?

Thanksgiving distracted me from this thread, but I wanted to reply to this before I forget. Assuming there is a significant distance between the GPS satellite and the person on the ground, so that you can't pick a window of spacetime containing both of them where spacetime curvature can be treated as negligible, then you're right that there's no equivalent to this in flat spacetime. From what I've read, all spacetime curvature is associated with some sort of measurable tidal effects; so this is why the equivalence principle only works in limits where curvature goes to zero, normally by zooming in on a small patch of spacetime, although I think you could also imagine letting the patch of spacetime have a fixed size but taking the limit as the size of the gravitating body goes to infinity.

 

[gonegahgah]

Cool.

I was wondering something. As you say you can have small windows of eqivalence.
Basically for a GPS satellite you have a weightless satellite and you have an observer with weight on the ground.
Would this be somehow equivalent in someway to someone being accelerated equivalent to surface g in deep space while a satellite traveled at the GPS speed?

I've just read what I wrote here and I can see the problem is that the person being accelerated would only have one instant where the satellite was traveling at GPS speed relative to them; the acceleration changing the relative speed constantly. So again there would only be a single window instant where this would be equivalent to the Earth scenario.
That's correct hey?

 

[JesseM]

Cool.

I was wondering something. As you say you can have small windows of eqivalence.
Basically for a GPS satellite you have a weightless satellite and you have an observer with weight on the ground.
Would this be somehow equivalent in someway to someone being accelerated equivalent to surface g in deep space while a satellite traveled at the GPS speed?

I've just read what I wrote here and I can see the problem is that the person being accelerated would only have one instant where the satellite was traveling at GPS speed relative to them; the acceleration changing the relative speed constantly. So again there would only be a single window instant where this would be equivalent to the Earth scenario.
That's correct hey?

I'm not sure there's any good way of defining "equivalence" such that a situation involving clocks with a large separation in curved spacetime (like the Earth clock and the GPS clock) can be called "equivalent" to any situation in SR, even for a moment. Normally "equivalence" refers to the idea that any experiment done in a certain patch of spacetime in the GR scenario will have the same result as the same experiment done in a certain patch of spacetime in an SR scenario, and even if the difference between the clock rates might be the same for the Earth clock and the GPS clock accelerating in flat spacetime as you envision, there could be other experiments done in this window which would not have the same result as they would in the curved spacetime around the Earth (perhaps one could take a large spring with one end near the Earth clock and one end near the GPS clock and see how it is pulled by tidal forces, for instance).