What's the underlying frame of the Einstein's Field Equation?

In summary, GR is generally covariant, meaning that the Einstein's Field Equation can be solved in any coordinate system. However, for more complicated solutions, a clever choice of coordinates is crucial. The ADM formalism is often used for solving the EFE, along with a (1+3)-formalism for numerical simulations. The metric of spacetime is not irrelevant, as it is the solution to the ten independent differential equations. Gravity is considered to warp spacetime through the metric, rather than being a force. In solving the EFE, it is often helpful to use an arbitrary map to simplify the equations before mapping it back to the original frame.
  • #141
Orodruin said:
his will not be the distance measured by the method suggested by the OP (see eg #123) - which was bouncing a light signal and measuring the round trip time - due to Shapiro delay.
Ah, yes, I didn't take not of the "by telemetry" in the post I was responding to. In principle there are ways of obtaining the radial distance by measurements, but it won't be anything as simple as just computing it directly from round-trip light travel times.
 
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  • #142
Orodruin said:
No. This will not be the distance measured by the method suggested by the OP (see eg #123) - which was bouncing a light signal and measuring the round trip time - due to Shapiro delay.
Does Shapiro delay basically take in account the displacement of the spaceship w.r.t. the star (same ##r, \phi## but different ## \theta##) during the process of sending and getting back the light signal off the orbiting spaceship to the star ?
 
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  • #143
cianfa72 said:
Does Shapiro delay basically take in account the displacement of the spaceship w.r.t. the star (same ##r## but different ##\theta, \phi##) during the process of sending and getting back the light signal off the orbiting spaceship to the star ?
Not in the form I mentioned, that assumed a stationary. Shapiro delay itself of course will exist regardless, but the general case will be more convoluted.
 
  • #144
Orodruin said:
Not in the form I mentioned, that assumed a stationary.
Do you mean a spaceship hovering w.r.t. the star at fixed ##r, \theta, \phi## ?
 
  • #145
cianfa72 said:
Do you mean a spaceship hovering w.r.t. the star at fixed ##r, \theta, \phi## ?
That is what stationary means, yes.
 
  • #146
Orodruin said:
That is what stationary means, yes.
ok, so in case of stationary spaceship (i.e. hovering at fixed ##r, \theta, \phi##) can we assume as "physical distance" from the star c times the average round-trip time for a bouncing light signal as measured from the spaceship ?
 
  • #147
cianfa72 said:
ok, so in case of stationary spaceship (i.e. hovering at fixed ##r, \theta, \phi##) can we assume as "physical distance" from the star c times the average round-trip time for a bouncing light signal as measured from the spaceship ?
No, as I mentioned in previous posts, ”distance” comes with a huge caveat in GR. What is typically defined as ”radial distance” is the integral
$$
\int \sqrt{g_{rr}} dr
$$
with appropriate boundary conditions. This corresponds to the distance along a hypersurface of constant ##t##-coordinate.

What would be measured by a radar measurement’s round trip time (multiplied by c/2) would be
$$
\int \sqrt{-\frac{g_{rr}}{g_{tt}}} dr = \int g_{rr} dr.
$$
 
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  • #148
Orodruin said:
What would be measured by a radar measurement’s round trip time (multiplied by c/2) would be
$$
\int \sqrt{\frac{g_{rr}}{g_{tt}}} dr = \int g_{rr} dr.
$$
ah, I believe we get this formula considering a lightlike geodesic starting from the event ##(r,\theta,\phi,t_0)##, bouncing at the center of the star and back to the event ##(r,\theta,\phi,t_1)##. Then from the difference ##\Delta t## we derive the interval of proper time ##\Delta \tau## elapsed at point ##r,\theta,\phi## and multiplying for c/2 we get the "distance" radar measurement.
 
  • #149
There is another adjustment to consider of course: The integrals give ##\Delta t##, but the locally measured time is of course gravitationally time dilated relative to this.
 
  • #150
Orodruin said:
There is another adjustment to consider of course: The integrals give ##\Delta t##, but the locally measured time is of course gravitationally time dilated relative to this.
I take it as for the lightlike path described in #148 the ##\Delta t## is given by the integral as in your #147. Then to get the proper time (since gravitational time dilation) that difference in coordinate time ##\Delta t## has to be rescaled accordingly.
 
  • #151
Btw is the curve ##(r, \theta_0, \phi_0,t_0)## with ##r## in the interval ##r_0-r_1## a geodesic of the spacelike hypersurface ##t=t_0## ?

In other words is the integral (i.e. the definition of "distance")
$$
\int \sqrt{g_{rr}} dr
$$ the spacetime length of a geodesic of the spacelike hypersurface ##t=0## ?
 
  • #152
Yes. It is relatively easy to conclude this from symmetry.
 
  • #153
Orodruin said:
What would be measured by a radar measurement’s round trip time (multiplied by c/2) would be
$$
\int \sqrt{-\frac{g_{rr}}{g_{tt}}} dr = \int g_{rr} dr.
$$
Orodruin said:
There is another adjustment to consider of course: The integrals give ##\Delta t##, but the locally measured time is of course gravitationally time dilated relative to this.
cianfa72 said:
I take it as for the lightlike path described in #148 the ##\Delta t## is given by the integral as in your #147. Then to get the proper time (since gravitational time dilation) that difference in coordinate time ##\Delta t## has to be rescaled accordingly.
Not so trivial to get ##\Delta t## from ##\Delta \tau## (your measured round-trip time) , since ##{\Delta \tau}/{\Delta t}## is not constant.
And once you somehow get your "measured" ##c\Delta t/2 = C(onstant)##, should you solve numerically the equation:
$$ C = \int^{r_1}_{r_0} g_{rr} dr $$
for ##r_1##, then plug it into the metric to evaluate it?
The ##r_{0}## value (coordinate radius of the neutron star) is also unknown BTW, maybe we could set it to 0 as a first approximation.
 
  • #154
Pyter said:
since Δτ/Δt is not constant.
Yes it is.

Pyter said:
The r0 value (coordinate radius of the neutron star) is also unknown BTW, maybe we could set it to 0 as a first approximation.
Definitely not. That would involve going below the horizon of the Schwarzschild metric where r is no longer space-like.

If you know the mass of the star, the easiest way to get the r coordinate would be to measure the proper acceleration required to remain stationary. (If you are stationary.)

Alternatively measure the proper time for a full orbit if you are in circular orbit.
 
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  • #155
Orodruin said:
Yes it is.
Doesn't also ##g_{tt}## depend on r?
Orodruin said:
If you know the mass of the star, the easiest way to get the r coordinate would be to measure the proper acceleration required to remain stationary. (If you are stationary.)

Alternatively measure the proper time for a full orbit if you are in circular orbit.
That would be equivalent to "measure" g**, then work backwards to obtain r. But if I already have determined g**, I don't need r.
 
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  • #156
Pyter said:
Doesn't also ##g_{tt}## depend on r?
Yes, but your r is fixed.
 
  • #157
Orodruin said:
Yes, but your r is fixed.
But that of the light/radio beam isn't.
Anyway, even if I should only consider my fixed r for computing ##\Delta t##, it means that also the LHS (C) of the equation in #139 depends on the unknown r.
 
  • #158
Pyter said:
But that of the light/radio beam isn't.
That is irrelevant, it is not what you would be measuring.

Pyter said:
it means that also the LHS (C) of the equation in #139 depends on the unknown r.
So what? Equations like ##f(x) = g(x)## can many times be solved.
 
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  • #159
As far as I can understand the round-trip time measurement in term of coordinate time ##\Delta t## from the hovering spaceship at coordinate radius ##r_1## from the star using a bouncing light beam, is actually the integral (divided by c/2)

$$
\int \sqrt{-\frac{g_{rr}}{g_{tt}}} dr = \int g_{rr} dr.
$$
evaluated between ##r_0## and ##r_1## (##r_0## is the coordinate radius of the star itself).
 
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  • #160
  • #161
So you need at least an estimate of M to compute ##r_0##, provided the density of neutronium is an invariant. But is it? I doubt it, because M certainly is, but the measured radius/volume of the star isn't.

EDIT: or better, the rest mass of the star is surely an invariant. The ##\gamma## factor changes with the observer, so the relativistic mass isn't.
 
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  • #162
Pyter said:
The ##r_{0}## value (coordinate radius of the neutron star) is also unknown BTW
You can measure ##r_0## by measuring the star's surface area or circumference and using the formulas ##C = 2 \pi r## or ##A = \pi r^2##.
 
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  • #163
PeterDonis said:
You can measure ##r_0## by measuring the star's surface area or circumference and using the formulas ##C = 2 \pi r## or ##A = \pi r^2##.
But the measured ##r_0## matches the coordinate ##r_0##, needed for the integral, only for the observer at infinity. You'd need at least another equation to link them.
 
  • #164
Pyter said:
But the measured ##r_0## matches the coordinate ##r_0##, needed for the integral, only for the observer at infinity. You'd need at least another equation to link them.
No. The ##r_0## measured in the way described (area of sphere or circumference of circle) is by definition the ##r## coordinate of the surface.
 
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  • #165
Pyter said:
But the measured ##r_0## matches the coordinate ##r_0##, needed for the integral, only for the observer at infinity.
Wrong. The ##r## coordinate is a coordinate. Coordinates are valid for any observer that uses that coordinate chart.
 
  • #166
PeterDonis said:
Wrong. The ##r## coordinate is a coordinate. Coordinates are valid for any observer that uses that coordinate chart.
Yes and that's just the point of these whole operations. To match the (absolute) coordinates with the (relative) physical distances measured by an arbitrary observer.
Orodruin said:
No. The ##r_0## measured in the way described (area of sphere or circumference of circle) is by definition the ##r## coordinate of the surface.
Now I guess that to send a probe with a rod to the surface of the star is out if discussion, unless we can prevent the tidal effects from tearing it to smithereens.
The only viable way to measure the apparent diameter could be from the visual angle covered by the star disc from a known measured distance, and from that compute ##r_0## using the metric.

EDIT: I'm also doubting that the observed ##M## is an absolute. Isn't that in the metric the mass of the star observed at infinity? Any other observer would have a ##\gamma <> 1 ## and thus see a different relativistic mass.
 
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  • #167
Pyter said:
The only viable way to measure the apparent diameter could be from the visual angle covered by the star disc from a known measured distance, and from that compute ##r_0## using the metric.
1) You could measure the gravitational field strength (and thereby map out a circle using points of equal local acceleration).

2) You could use local light rays to map out an n-gon with a large number of sides. That would be a very good approximation of a circle.

That would also give you an estimate for the mass of the star.

PS 3) Something clever using gyroscopes.
 
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  • #168
Pyter said:
Yes and that's just the point of these whole operations. To match the (absolute) coordinates with the (relative) physical distances measured by an arbitrary observer.
That doesn't change the fact that your claim in post #163 was wrong, as I responded. The ##r## coordinate in Schwarzschild coordinates does directly represent tangential distances (i.e., distances around a circle at that radial coordinate); that's obvious from its definition. It just doesn't directly represent radial distances.

Pyter said:
Now I guess that to send a probe with a rod to the surface of the star is out if discussion, unless we can prevent the tidal effects from tearing it to smithereens.
This is a thought experiment, so you can do anything that's consistent with the laws of physics. The laws of physics don't forbid building a probe that can withstand being on a neutron star's surface.

Pyter said:
The only viable way to measure the apparent diameter could be from the visual angle covered by the star disc from a known measured distance, and from that compute ##r_0## using the metric.
This is one way, but by no means the only viable way. See above.

Pyter said:
EDIT: I'm also doubting that the observed ##M## is an absolute. Isn't that in the metric the mass of the star observed at infinity?
No. It's an invariant parameter in the metric, which is equal to the star's rest mass. Any observer can measure it in a number of different ways; for example, by putting a test object in orbit around the mass (the "test object" could be the observer's own spaceship) and measuring its orbital parameters.

Pyter said:
Any other observer would have a ##\gamma <> 1 ## and thus see a different relativistic mass.
The ##M## in the metric is not relativistic mass. See above.
 
  • #169
@Pyter, you keep making intuitive guesses that are wrong. You should realize by now that your intuition is flawed and stop doing that.
 
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  • #170
Pyter said:
EDIT: I'm also doubting that the observed ##M## is an absolute. Isn't that in the metric the mass of the star observed at infinity? Any other observer would have a ##\gamma <> 1 ## and thus see a different relativistic mass.
If you are still using relativistic mass, then I'm not sure how much sense GR is going to make.
 
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  • #172
PeterDonis said:
That doesn't change the fact that your claim in post #163 was wrong, as I responded. The ##r## coordinate in Schwarzschild coordinates does directly represent tangential distances (i.e., distances around a circle at that radial coordinate); that's obvious from its definition. It just doesn't directly represent radial distances.
Ok, you're measuring the effective coordinate through the measured circumference/area.
PeterDonis said:
This is a thought experiment, so you can do anything that's consistent with the laws of physics. The laws of physics don't forbid building a probe that can withstand being on a neutron star's surface.
I was putting myself in the shoes of a spaceship pilot who needs to compute the metric tensor in practice, for astrogation purposes.
PeterDonis said:
No. It's an invariant parameter in the metric, which is equal to the star's rest mass. Any observer can measure it in a number of different ways; for example, by putting a test object in orbit around the mass (the "test object" could be the observer's own spaceship) and measuring its orbital parameters.

The ##M## in the metric is not relativistic mass. See above.
Who said the contrary? I was talking about the observed mass from some arbitrary point. Isn't the rest mass equal to the mass observed at infinity, where the space is flat?
PeterDonis said:
@Pyter, you keep making intuitive guesses that are wrong. You should realize by now that your intuition is flawed and stop doing that.
I partially disagree, some of them proved right. I don't pretend to be right 100% of the times anyway, I'm here to learn. Thanks for your corrections.
 
  • #173
Pyter said:
Isn't the rest mass equal to the mass observed at infinity, where the space is flat?
It’s equal to the mass observed anywhere else too…
 
  • #174
Pyter said:
you're measuring the effective coordinate through the measured circumference/area.
It's not an "effective coordinate". It's the ##r## coordinate in the coordinate chart you've defined. There is no "effective" about it.

Pyter said:
Isn't the rest mass equal to the mass observed at infinity, where the space is flat?
No, it's equal to the rest mass, period. There is no such thing as "rest mass observed at a particular point" that changes from point to point or observer to observer. Rest mass is an invariant.
 
  • #175
It is very simple to remember, what's measured by a given local observer in GR. Coordinates are arbitrary "maps" of spacetime points, which do not have a priori any physical meaning, because they change under general (local) coordinate transformations (local diffeomorphism invariance), and thus this is a gauge symmetry, and that's why the Einstein field equations for the pseudometric are only solvable modulo these general coordinate transformations.

To know what a local observer measures, you have to use local scalar quantities referring to this observer. Formally the local observer moves along a time-like world line. Then you can define a tetrad along this world line for this observer by choosing at an arbitrary point at this world line the time-like normalized tangent, pointing to the future, to the world line and then three space-like Minkowski-orthonormal vectors. Then you Fermi-Walker transport this tetrad further along the worldline to have a locally non-rotating local inertial frame for this observer. Then all local observables can be defined wrt. this reference frame as in special relativity.
 
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