SR-time dilation vs GR-time dilation on rotating Earth

[cianfa72]

But also from the discussion in this thread, the error involved in treating the ECI frame Cartesian coordinates as if they were Euclidean, i.e., treating the metric on spacelike hypersurfaces of constant time as if it were flat, is negligible for practical purposes. That's why Ashby does so in the paper.

So his locally in "locally inertial frame whose origin is attached to the center of the earth" actually spatially extends from the Earth's center up to locations of GPS satellites orbits. Furthermore, due to the constancy of light speed in ECI frame, clocks at rest in it can be Einstein's synchronized consistently.

 

[PeterDonis]

So his locally in "locally inertial frame whose origin is attached to the center of the earth" actually spatially extends from the Earth's center up to locations of GPS satellites orbits.

Yes.

due to the constancy of light speed in ECI frame, clocks at rest in it can be Einstein's synchronized consistently.

Clocks at rest, yes. But no clocks that are in common use by anyone on Earth are at rest in the ECI frame. They are either at rest on the rotating Earth, meaning they are moving in the ECI frame at Earth's rotation speed, or they are in free-fall orbits, like the GPS satellite clocks, and are moving in the ECI frame at free-fall orbital speed.

 

[Vanadium 50]

At the risk of derailing the thread by actually discussing the problem:

1. Can one treat the gravitational time dilation from the earth as a geoid? And how different from a sphere is this?
2, A body in orbit an inch above the Earth's surface will experience, a) about the same time dilation as from gravity, b) many, many times more, or c) many, many times less?
3. An object at rest on the Earth is going, a) near orbital speed, b) much faster than orbital speed, ir c) much slower than orbital speed.

Now you have everything you need to solve your problem. No Kerr Metric, no odd coordinate systems, nothing bit algebra needed.

 

[cianfa72]

Coming back to the main point: in the approximation given by Ashby's paper the spacetime around the Earth (for pratical purposes in a sufficient large spatial volume around it) is assumed to be flat (Minkowskian). However in ECI frame there exists a gravitational potential (alike the Newton gravitational potential in inertial coordinates). This potential and the kinematic motion of the points on the geoid (i.e. the geoid's timelike worldlines which are members of its worldtube) in ECI frame give rise to an effective potential in that frame which is "responsable" for the shape of the geoid itself.

 

[PeterDonis]

in the approximation given by Ashby's paper the spacetime around the Earth (for pratical purposes in a sufficient large spatial volume around it) is assumed to be flat (Minkowskian).

Wrong. The spacelike surfaces of constant coordinate time are assumed to be flat. That is not the same as the spacetime being flat.

 

[cianfa72]

Wrong. The spacelike surfaces of constant coordinate time are assumed to be flat. That is not the same as the spacetime being flat.

Ah ok, therefore, as far as I understand, Ashby's paper assumes a stationary spacetime. The coordinate time $t$ is "adapted" to the timelike KVF (i.e. the vector field $\partial_t$ is the timelike KVF). Since spacelike hypersurfaces of constant coordinate time $t$ are assumed to be flat (Euclidean), one can pick a set of timelike worldlines filling the spacetime such that, on each of those spacelike hypersurface, they define cartesian coordinates "at rest" in ECI frame.

 

[PeterDonis]

Ashby's paper assumes a stationary spacetime.

Yes.

The coordinate time $t$ is "adapted" to the timelike KVF

Yes.

Since spacelike hypersurfaces of constant coordinate time $t$ are assumed to be flat (Euclidean), one can pick a set of timelike worldlines filling the spacetime such that, on each of those spacelike hypersurface, they define cartesian coordinates "at rest" in ECI frame.

They define the ECI frame itself. It makes no sense to say coordinates are "at rest in the ECI frame". The coordinates are the frame.

 

[cianfa72]

They define the ECI frame itself. It makes no sense to say coordinates are "at rest in the ECI frame". The coordinates are the frame.

Yes, I was sloppy. I meant that within the congruence of timelike worldlines that define the ECI frame (by definition they are "at rest" in the frame being defined), one can pick specific members for the "coordinates grid" - Cartesian coordinates with origin in the Earth's center and $x,y,z$ axes as described in post #64 for instance.

 

[PeterDonis]

within the congruence of timelike worldlines that define the ECI frame

The worldlines by themselves don't define a complete frame. Only the worldlines plus a particular choice of spatial coordinates to label them defines a unique frame. You have already been given references for how the ECI frame is defined.

 

[PeterDonis]

At this point the OP question has been well answered. Thread closed.