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

[cianfa72]

No, that's not quite correct either. The gravitational potential that is used in the ECI frame includes the effects of Earth's quadrupole moment, so it recognizes that the Earth is not precisely spherical. This affects the definition of the geoid.

Ok, so ECI coordinates are not simply the "inverse mapping" of spatial Schwarzschild coordinates $(r,\theta,\phi)$ of Schwarzschild spacetime.

Not quite, no. The rotation parameter $a$ does not vanish at infinity, so the coordinates cannot be standard polar coordinates.

Ah ok, the Kerr metric is asymptotically flat however the spatial BL-coordinates at infinity are not the standard polar coordinates.

 

[PeterDonis]

so ECI coordinates are not simply the "inverse mapping" of spatial Schwarzschild coordinates $(r,\theta,\phi)$ of Schwarzschild spacetime.

That's correct.

the Kerr metric is asymptotically flat however the spatial BL-coordinates at infinity are not the standard polar coordinates.

That's correct.

 

[cianfa72]

A meaningful question would be whether a worldline with fixed spatial BL coordinates in Kerr spacetime is at rest relative to an observer at rest at infinity. The answer to this should be simple to read off from the Kerr metric and its limit as $r \to \infty$.

About this point I reasoned as follows: since the Kerr metric is stationary it follows that light beam paths $ds^2=0$ with fixed $(\theta, \phi)$ are described in BL-coordinates by ${dr} / {dt}$ that doesn't depend on coordinate time $t$ along the path. Hence light beam paths emitted at different coordinate time from an observer at fixed $(r,\theta, \phi)$ -- included at $r \to \infty$ -- are actually "congruent" in BL-chart. Therefore the amount of coordinate time between sending and receiving back a light beam is always the same. Since the metric doesn't depend on $t$ the amount of proper time along the observer worldline doesn't change as well.

 

[PeterDonis]

I reasoned as follows

Yes, this is correct.

 

[cianfa72]

Yes, this is correct.

Ok, therefore the above reasoning holds true in any stationary spacetime.

 

[PeterDonis]

Ok, therefore the above reasoning holds true in any stationary spacetime.

As you stated it, it only holds for radial trajectories in a stationary spacetime in which spherical spatial coordinates can be adopted.

However, it should be straightforward to see that the argument can be generalized so as to apply to any round-trip light signals between two fixed integral curves of the timelike Killing vector field.

 

[cianfa72]

However, it should be straightforward to see that the argument can be generalized so as to apply to any round-trip light signals between two fixed integral curves of the timelike Killing vector field.

Yes, since in a stationary spacetime there are coordinates such that integral curves of timelike KVF are described by fixed spatial (spacelike) coordinates and varying coordinate time $t$. Therefore, fixed two of them, one can apply the above argument to get the result.

 

[PeterDonis]

fixed two of them, one can apply the above argument to get the result.

You don't even have to use coordinates to make the argument. Just the fact that you have two fixed integral curves of the timelike KVF is enough.

 

[cianfa72]

You don't even have to use coordinates to make the argument. Just the fact that you have two fixed integral curves of the timelike KVF is enough.

Ok, the point is that light cones from any point on fixed integral curves of timelike KVF are congruent each other (when "trasported" along the flow of timelike KVF).

 

[PeterDonis]

light cones from any point on fixed integral curves of timelike KVF are congruent each other (when "trasported" along the flow of timelike KVF).

Yes.

 

[cianfa72]

Coming back to ECI frame: we said it is not the "inverse mapping" of spherical coordinates of Schwarzschild spacetime in Schwarzschild coordinates. So which is their significance ?

In other word: which is the "rule" for assigning ECI coordinates to a point ?

 

[PeterDonis]

which is the "rule" for assigning ECI coordinates to a point ?

https://en.wikipedia.org/wiki/Earth-centered_inertial

 

[cianfa72]

The Wikipedia link claims that for ECI frame Cartesian coordinates can be used. However my doubt is: since spacetime is not flat, $z$ coordinate of a point along the Earth's rotating axis is not the Euclidean length from the Earth's center up to that point P along $z$ axis. In other words Euclidean geometry doesn't apply to such Cartesian coordinates.

 

[PeterDonis]

The Wikipedia link claims that for ECI frame Cartesian coordinates can be used.

Yes.

However my doubt is: since spacetime is not flat

Cartesian coordinates do not require spacetime to be flat. Cartesian coordinates does not mean the metric of a spacelike hypersurface of constant time must be Euclidean, any more than spherical coordinates means $r$ must be the physical distance from the center.

 

[cianfa72]

Cartesian coordinates do not require spacetime to be flat. Cartesian coordinates does not mean the metric of a spacelike hypersurface of constant time must be Euclidean, any more than spherical coordinates means $r$ must be the physical distance from the center.

Ok, so the key feature of Cartesian coordinates is that they are composed by mutually orthogonal geodesic coordinate lines regardless of the fact that coordinate values may not be the same as the physical distance measured along any of them from the center.

 

[PeterDonis]

the key feature of Cartesian coordinates is that they are composed by mutually orthogonal geodesic coordinate lines regardless of the fact that coordinate values may not be the same as the physical distance measured along any of them from the center.

Yes.

 

[cianfa72]

The weird thing is that, for example in 2D cartesian coordinates, a geodesic starting orthogonal to the $x$ axis and another starting orthogonal to the $y$ axis might not intersect orthogonally (i.e. forming 4 congruent angles).

 

[PeterDonis]

in 2D cartesian coordinates

On what manifold? A flat Euclidean plane? Or something else?

 

[cianfa72]

On what manifold? A flat Euclidean plane? Or something else?

I was talking about a generic 2D surface (or better a non-Euclidean 3D space), like the spacelike hypersurface of constant coordinate time $t$ discussed in post #49.

 

[PeterDonis]

I was talking about a generic 2D surface (or better a non-Euclidean 3D space)

Cartesian coordinates are generally used for cases where the space is very close to Euclidean. That is true for the spacelike surfaces of constant time that you refer to.

 

[cianfa72]

Cartesian coordinates are generally used for cases where the space is very close to Euclidean. That is true for the spacelike surfaces of constant time that you refer to.

So, as in Schwarzschild spacetime in Schwarzschild coordinates the coordinate $r$ represents the square root of nested 2-spheres's area divided by $4\pi$ (and it is not the radial distance from the center along the spacelike geodesic integral curves of $\partial_r$), which is the significance of Cartesian coordinates (e.g. $x$) in ECI frame ?

 

[PeterDonis]

which is the significance of Cartesian coordinates (e.g. $x$) in ECI frame ?

Have you read the article I referenced? What does it say?

 

[cianfa72]

Have you read the article I referenced? What does it say?

Yes, the Wikipedia link says that the orientation of ECI frame is defined by the ecliptic and the Earth's rotation axis (i.e. the fundamental plane is the ecliptic).

The location of an object in space can be defined in terms of right ascension and declination which are measured from the vernal equinox and the celestial equator.

Then it claims:

Locations of objects in space can also be represented using Cartesian coordinates in an ECI frame.

However, it seems to me, there is not a definition of Cartesian coordinates applied to ECI frame. Are they just the "inverse mapping" of right ascension and declination in spherical coordinates ?

 

[PeterDonis]

Are they just the "inverse mapping" of right ascension and declination in spherical coordinates ?

That in itself is not sufficient; you would also need $r$ in spherical coordinates. The vernal equinox and the celestial equator define the orientation of the Cartesian axes.

 

[cianfa72]

That in itself is not sufficient; you would also need $r$ in spherical coordinates. The vernal equinox and the celestial equator define the orientation of the Cartesian axes.

Yes of course. For any 2-sphere of given $r$ of where the point to be located is, one gets the Cartesian coordinates by "inverse mapping" its right ascension and declination.

So the question is: what is the $r$ coordinate ? I don't think it is the $r$ coordinate of Schwarzschild coordinates in Schwarzschild spacetime (i.e. it doesn't have the same significance/feature as "reduced circumference").

 

[PeterDonis]

the question is: what is the $r$ coordinate ? I don't think it is the $r$ coordinate of Schwarzschild coordinates in Schwarzschild spacetime (i.e. it doesn't have the same significance/feature as "reduced circumference").

I believe the usual radial coordinate for frames like the ECI frame is the isotropic radial coordinate, not the Schwarzschild radial coordinate--i.e., the coordinate is defined so that locally, radial coordinate increments are the same physical distance as tangential coordinate increments (i.e., $dr$ vs. $r d\theta$ or $r d \phi$).

However, in the weak field, slow motion approximation, it doesn't really matter; the Schwarzschild and isotropic radial coordinates both have the same weak field, slow motion limit. In that limit, spatial coordinate increments are as if the space was Euclidean, i.e., the difference is too small to matter when compared with the difference in increments of the time coordinate vs. proper time.

 

[cianfa72]

In that limit, spatial coordinate increments are as if the space was Euclidean

You mean Cartesian spatial coordinate increments (in ECI frame) are as if the space was Euclidean.

i.e., the difference is too small to matter when compared with the difference in increments of the time coordinate vs. proper time.

You mean the difference in increments of time coordinate vs. proper time (along the integral curves of $\partial_t$) is much more large than the difference/discrepancy between Cartesian spatial increments vs. proper distance (along the integral curves of $\partial_x$, $\partial_y$ or $\partial_z$).

 

[PeterDonis]

You mean Cartesian spatial coordinate increments (in ECI frame) are as if the space was Euclidean.

Euclidean space can be described by coordinates other than Cartesian. In spherical coordinates the relevant increments, as I've already posted once, would be $dr$, $r d\theta$, and $r d \phi$ (actually the last is only correct in the equatorial plane, elsewhere it would be $r \sin \theta d \phi$).

You mean the difference in increments of time coordinate vs. proper time (along the integral curves of $\partial_t$) is much more large than the difference/discrepancy between Cartesian spatial increments vs. proper distance (along the integral curves of $\partial_x$, $\partial_y$ or $\partial_z$).

Yes.

 

[cianfa72]

Euclidean space can be described by coordinates other than Cartesian. In spherical coordinates the relevant increments, as I've already posted once, would be $dr$, $r d\theta$, and $r d \phi$ (actually the last is only correct in the equatorial plane, elsewhere it would be $r \sin \theta d \phi$).

Ok therefore, even though the spacelike hypersurfaces of constant coordinate time $t$ are not Euclidean around the Earth, we can use in ECI frame Cartesian coordinates centered on Earth's center (with $x$,$y$ axes on ecliptic plane oriented along the vernal equinox and celestial equator and $z$ along the Earth's rotating axis) given as "inverse mapping" of $(r,\theta, \phi)$ where $r$ is the isotropic radial coordinate. It is defined such that locally the proper distance for increment $dr$ along $\partial_r$ is the same as proper distance for increments $rd\theta$ and $r\sin \theta d\phi$ along $\partial_{\theta}$ and $\partial_{\phi}$ respectively.

 

[PeterDonis]

Ok therefore, even though the spacelike hypersurfaces of constant coordinate time $t$ are not Euclidean around the Earth, we can use in ECI frame Cartesian coordinates centered on Earth's center (with $x$,$y$ axes on ecliptic plane oriented along the vernal equinox and celestial equator and $z$ along the Earth's rotating axis) given as "inverse mapping" of $(r,\theta, \phi)$ where $r$ is the isotropic radial coordinate. It is defined such that locally the proper distance for increment $dr$ along $\partial_r$ is the same as proper distance for increments $rd\theta$ and $r\sin \theta d\phi$ along $\partial_{\theta}$ and $\partial_{\phi}$ respectively.

All this is just repeating things that have already been said.

 

[Vanadium 50]

All this is just repeating things that have already been said.

And avoiding the original question.
And avoiding the calculation.

 

[pervect]

The first step of doing a calculation is finding a metric. The most realistic choice is NOT going to be Kerr or Schwarzschild, because the Earth is not an idealized black hole. Since the OP is still asking about which metric to use, I don't think they are prepared to do a calculation as of yet, and that if they think about the steps they need to do a calculation, the first one is to find a metric.

I'm pretty sure the OP has at least seen Neil Ashby's rather famous paper on relativity in the GPS, for instance https://www.aapt.org/doorway/tgru/articles/ashbyarticle.pdf. But really, there are a lot of possible choices, and it's the OP's job to make one. My impression is that they are struggling to conceptually answer the question of what metric to use, I can and have made some suggestions that might be interesting.

So the first thing I would point out is that this paper does (IIRC) give some suggestions as to a metric.

The ones I would suggest are Ashby's paper (not my favorite, but the OP uses a lot of language from it, so I think it's a paper that they've read that may be important to their thought process), Misner's "Precis of General Relativity" which is basically a response to Ashby's paper (I prefer Misner's approach over Ashby's, it's at least worth reading). I'm also partial to MTW's treatment of PPN theory in "Gravitation".

Honoroable mention should be given to the IAU's approach, as a highly accurate one that's of obvious official interest, but it is probably not the best place to start. This has several revisions, so one would have to pick one.

As an overview: Basically rather than using an exact analytical solution that doesn't fit the actual Earth , such as Schwarzschild or Kerr, I would use some model based on the linearized gravity.

After one has picked one (or more) metrics of interest that one believes model the metric of space-time around the Earth, then one is in a position to do one or more calculations and compare the results and look at the significance of various effects.

I'm not quite sure why the OP is so focussed on coordinates, it's rather like they know that methods independent of a particular coordinate chart exist but they've forgotten this? Anyway, this is all a bit of a reahsh, unsuprisingly.

I'd also suggest looking at what we know about the Earth, for instance https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html, and suggest that some basic parameters of interest are GM, and J2. A brief theoretical diversion into into "potential theory" (see Goldstein's classical mechanics, the chapter on the Earth-moon system and the figure of the Earth) might help explain why J2, a ratio of moments of particular interest.

Anyway, I'm not holding my breath, but maybe we can move past "Should I use Schwarzschild or Kerr" :).

 

[Vanadium 50]

The first step of doing a calculation is finding a metric.t

I am going to disagree.

The first step is to write down what you want to calculate - e.g. the local time that elapses at the equator for each second that elapses at the poles. Δt/t equals something.

I would also suggest that the next step is to try and avoid a metric. Can you solve this by combining solutions you already know? This was suggested - and ignored - around 50 messages ago.

I'm not quite sure why the OP is so focused on coordinate

More generally, I am not sure why he has no apparent focus on actually solving the problem - a problem he himself posted. He's posted a lot of messages, and I struggle to find evidence that he can work any problem in GR at all. Maybe not even SR. That, of course, makes it difficult to discuss more difficult problems.

Let me add one more thing to your list - decide how good your answer has to be. There is zero point to worrying about part per trillion nonlinearities if you know where your clock is to a few meters.

 

[cianfa72]

I'm pretty sure the OP has at least seen Neil Ashby's rather famous paper on relativity in the GPS, for instance https://www.aapt.org/doorway/tgru/articles/ashbyarticle.pdf. But really, there are a lot of possible choices, and it's the OP's job to make one. My impression is that they are struggling to conceptually answer the question of what metric to use, I can and have made some suggestions that might be interesting

I read that paper. In section II he defines the ECI frame as

locally inertial frame whose origin is attached to the center of the earth but which is not rotating. This is called the “Earth-Centered Inertial,” or ECI frame.

So the Earth's center is at rest at the ECI frame's origin while it free-falls towards the Sun. He claims that the effect of the Sun (including other solar system bodies) comes in only through tidal forces (i.e. tidal gravitational potential) which, from the point of view of satellite's GPS clocks, can be disregared since negligible.

According to him light propagation in ECI frame takes place with constant speed $c$ in any direction. He then writes 4 equations describing in ECI frame the propagation of light signals from 4 GPS satellites up to a terrestrial receiver. Each equation is basically the equation for light propagation in standard inertial coordinates as if spacetime were flat (Minkowski).

From the discussion in this thread, actually Cartesian coordinates for ECI frame are not Euclidean in the sense that the metric on spacelike hypersurfaces of constant time is not Euclidean (even though the spacetime can be assumed to be stationary).

 

[PeterDonis]

From the discussion in this thread, actually Cartesian coordinates for ECI frame are not Euclidean in the sense that the metric on spacelike hypersurfaces of constant time is not Euclidean

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.