I Speed of light not an invariant in GR

[Pyter]

As explained, I used $ds$ to be the spacelike interval traveled by the light. To avoid confusion, we could use $dl$ for this. The calculation in flat spacetime (in some inertial frame) would look like:

[...]

3) The spatial distance traveled by the light (as measured in the IRF) is $\Delta l^2 = (x_1 - x_0)^2 + (y_1 - y_0)^2 + (z_1 - z_0)^2$.

At last. You do need a measured Euclidean length at the numerator to compute a velocity.

@vanhees71 it's all clear (more or less) except this point:

The time measured by his clock is his "proper time" and given in the used coordinates by
$$\mathrm{d} \tau^2=\mathrm{d} s^2=g_{00} (\mathrm{d} x^0)^2.$$
I used a natural system of units where $c=1$.

As you say, the time measured by his clock is his "proper time", which as we all know is also designated by the symbol $\tau$. But this time (difference) is $\mathrm{d}x^0$ (his local time coordinate), and in general $g_{00} \neq 0$, so how's possible that $\mathrm{d}\tau^2 =( \mathrm{d}x^0)^2$?

 

[PeterDonis]

You do need a measured Euclidean length at the numerator to compute a velocity.

No. You need a measured length. But, as I have already pointed out several times now, the length does not have to be "Euclidean". It only happens to be in the particular case @PeroK gave because he is assuming flat spacetime. In a general curved spacetime the length would not be "Euclidean" and its functional dependence on the coordinate differences would be more complicated.

 

[PeterDonis]

this time (difference) is $\mathrm{d}x^0$ (his local time coordinate)

Only if $g_{00} = 1$. But as you note, in general that is not the case. If $g_{00} \neq 1$ then $\mathrm{d}\tau$ is not the same as $\mathrm{d}x^0$. (You and @vanhees71 are also assuming that the observer is at rest in the given coordinates, so only $\mathrm{d}x^0$ is nonzero.)

how's possible that $\mathrm{d}\tau^2 =( \mathrm{d}x^0)^2$?

It doesn't in general. Read the formula @vanhees71 actually wrote more carefully.

 

[Pyter]

Only if g00=1. But as you note, in general that is not the case. If g00≠1 then dτ is not the same as dx0. (You and @vanhees71 are also assuming that the observer is at rest in the given coordinates, so only dx0 is nonzero.)

It doesn't in general. Read the formula @vanhees71 actually wrote more carefully.

That's what I meant. How can it be that: $$\mathrm{d}\tau^ \triangleq \mathrm{d}x^0$$ (by definition) and: $$\mathrm{d} \tau^2=\mathrm{d} s^2=g_{00} (\mathrm{d} x^0)^2$$ at the same time, if $g_{00} \neq 1$?

 

[PeroK]

That's what I meant. How can it be that: $$\mathrm{d}\tau^ \triangleq \mathrm{d}x^0$$ (by definition).

You could say that $dt = dx^0$, in the sense that $t = x^0$ is the "time" coordinate. And, of course, you have $\tau = t$ in a local inertial frame.

 

[Pyter]

@PeroK isn't the the proper time $\tau$, by definition, the time $x^0$ measured by an observer at the origin of the coordinate system where he's stationary?

 

[PeroK]

@PeroK isn't the the proper time $\tau$, by definition, the time $x^0$ measured by an observer at the origin of the coordinate system where he's stationary?

No.

 

[vanhees71]

At last. You do need a measured Euclidean length at the numerator to compute a velocity.

@vanhees71 it's all clear (more or less) except this point:As you say, the time measured by his clock is his "proper time", which as we all know is also designated by the symbol $\tau$. But this time (difference) is $\mathrm{d}x^0$ (his local time coordinate), and in general $g_{00} \neq 0$, so how's possible that $\mathrm{d}\tau^2 =( \mathrm{d}x^0)^2$?

Again you have to be careful, about which world line we are talking. For the world line of the light signal ("photon") you have $\mathrm{d} s^2=0$. For the world line of the observer at rest in the reference frame defined by the coordinates $x^{\mu}$ it's $\mathrm{d}s^2=g_{00} (\mathrm{d} x^0)^2$, and that's the proper time measured by the observer's clock, i.e., $\mathrm{d} \tau=\sqrt{g_{00}} \mathrm{d} x^0$.

One should be aware that all that's measurable by some device is an invariant quantity at a point, i.e., time intervals shown by a clock are proper times of some observer.

 

[Pyter]

I've checked out the proper time definition on Wikipedia and there are indeed two different definitions for SR and GR. What do you know, the definition for GR:
$$\Delta\tau = \int_P \, d\tau = \int_P \frac{1}{c}\sqrt{g_{\mu\nu} \; dx^\mu \; dx^\nu}$$
is claimed to be invariant and also contains c.
So we may safely assume, according to Wikipedia, that the postulate of c invariance also holds for GR.

 

[vanhees71]

As the name suggests SR is a special case of GR (describing situations, where gravitational fields can be neglected). The definition of proper time is the same in SR and GR. It's a functional of a time-like worldline and defined as written in Wikipedia for both SR and GR. The distinction is that in SR you can always introduce a global (!) inertial reference frame with a constant tetrad everywhere as reference frame. In this frame, globally you have $g_{\mu \nu}=\eta_{\mu \nu}$. If a gravitational field is present, you can only introduce coordinates at one point, such that $g_{\mu \nu}=\eta_{\mu \nu}$ at this one point but not globally!

You can understand GR as arising from SR by "gauging Lorentz symmetry", i.e., making Lorentz invariance a local symmetry. This is the modern definition of what's called "equivalence principle", i.e., it says that at any space-time point there is a local (!) inertial reference frame. If the gravitational field cannot be neglected, i.e., if the Riemann curvature tensor doesn't vanish, in a spacetime region, you never have a global inertial reference frame.

 

[PeroK]

I've checked out the proper time definition on Wikipedia and there are indeed two different definitions for SR and GR. What do you know, the definition for GR:
$$\Delta\tau = \int_P \, d\tau = \int_P \frac{1}{c}\sqrt{g_{\mu\nu} \; dx^\mu \; dx^\nu}$$
is claimed to be invariant and also contains c.
So we may safely assume, according to Wikipedia, that the postulate of c invariance also holds for GR.

The underlying postulate of both is that spacetime is a 4D manifold, characterised by some metric tensor, $g_{\mu\nu}$. The (mathematical) theory of manifolds tells you that the above integral is independent of your choice of coordinates. I.e. it is an invariant quantity.

We physically interpret that quantity as the proper time (along a timelike curve in the manifold). And we expect that quantity to be the time recorded by a clock that moves along that timelike curve. This is something we can test - especially for SR where we postulate that the spacetime is (at least approximately) flat.

Note that when we say the time measured by a clock, we really mean the time that passes for physical processes to evolve. So, we have a theory that tells us how much physical change will take place along any given timelike path. And when we test this (e.g. lifetime of high-speed particles as measured in the lab) we find the experiment matches the theory.

We have a further postulate that light moves on null curves. This can be tested (e.g. by measuring the deflection of light during a solar eclipse). This postulate, moreover, implies that a) in flat spacetime the speed of light is invariant - i.e. always measured as $c$ in an IRF; and b) in curved spacetime, for sufficiently local trajectories, the speed of light is measured locally as $c$.

Note that b) is easy to show for an observer at rest and a static, diagonal metric (which is the exercise I suggested you undertake).

 

[Pyter]

b) in curved spacetime, for sufficiently local trajectories, the speed of light is measured locally as .

What could be a non-local experiment where the light speed is not measured as c?

 

[PeroK]

What could be a non-local experiment where the light speed is not measured as c?

Consider a radial light ray in Schwarzschild coordinates: starting from some coordinate $R_0$ moving out to $R_1$, off a mirror and back again.

We can measure the proper distance from $R_0$ to $R_1$ by integrating the line element. This will gives some length $l$. The speed of light, as measured by an observer at $R_0$, will be $\frac{2l}{\Delta \tau_0}$.

To save the calculations, we can argue that this is not equal to $1$ as follows. The locally measured speed of light at each point is $1$, which means the rate of change $\frac{dl}{d\tau_r} = 1$. But, by relating $\tau_0$ to $\tau_r$ using the coordinate time $t$, we can see that $d\tau_r > d\tau_0$ for every point $r > R_0$ and hence $\frac{dl}{d\tau_0} > 1$. And we must end up with $\frac{2l}{\Delta \tau_0} > 1$.

An alternative argument is to compare the measured (round trip) speed of light for observers at $R_0$ and $R_1$. The proper length is the same and the coordinate time interval must be the same, but $\Delta \tau_0 \ne \Delta \tau_1$, so the measured speed of light must be different for those two observers.

Note: the previous calculations have argued that $$\lim_{l \rightarrow 0} \frac{2l}{\Delta \tau_0} = \lim_{l \rightarrow 0} \frac{2l}{\Delta \tau_1} = 1$$ And, in the example above, the measured speeds will approach the limit from above and below $1$ for $R_0$ and $R_1$ respectively - and only the limits are equal to $1$.

 

[Pyter]

Fascinating. c > 1 in some cases, that must be the famous warp speed. And the c appearing in the proper time's equation is just the limit as measured in an IFR, then.

 

[PeroK]

Fascinating. c > 1 in some cases, that must be the famous warp speed. And the c appearing in the proper time's equation is just the limit as measured in an IFR, then.

One of the things that GR requires is an ability to define things properly and organise your calculations.

For example, someone at rest in Schwarzschild coordinates is not moving inertially. GR is a subject that requires solid prerequisites, focus and disciplined thinking. If you don't have those, then you cannot make progress. You'll just end up wandering around the subject in a state of permanent confusion.

 

[Pyter]

@PeroK what did I say?

 

[PeroK]

@PeroK what did I say?

You said this:

Fascinating. c > 1 in some cases, that must be the famous warp speed. And the c appearing in the proper time's equation is just the limit as measured in an IFR, then.

Three statements, three errors.

 

[Pyter]

@PeroK you wrote earlier that in the radial case, on the long distance you measure c > 1. I assumed that's because the space is "contracted" radially and the light appears to move faster.
And what the c in the Wikipedia's proper time equation would be if not the one measured in a local IFR?

 

[PeroK]

@PeroK you wrote earlier that in the radial case, on the long distance you measure c > 1. I assumed that's because the space is "contracted" radially and the light appears to move faster.
And what the c in the Wikipedia's proper time equation would be if not the one measured in a local IFR?

I never wrote $c > 1$. I said that the measured (non-local) speed of light $> 1$. In what I did $c = 1$ by definition. In GR, $c$ is not really the speed of light but the conversion factor between units of length and time. The locally measured speed of light $= c$. Which is a consequence of light traveling on null paths.

If you are studying GR, you need to stop thinking in terms of IRF's all the time. Think coordinates, Schwarzschild or otherwise.

Also, you are doing none of the calculations yourself - and are misunderstanding all our posts and calculations.

You need to get a textbook and start posting the exercises as homework if you get stuck.

 

[Pyter]

@PeroK, you're right. Please read "the measured speed of light" instead of c in my previous post.
And no, actually it's not warp speed, because if it was the case, the observer in $R_1$ would observe it too, but he measures a speed of light < 1 instead.

 

[Pyter]

Thank you all for all the pointers and explanations.

 

[PeterDonis]

That's what I meant. How can it be that: $$\mathrm{d}\tau^ \triangleq \mathrm{d}x^0$$ (by definition)

There is no such definition. This statement is simply wrong whenever $g_{00} \neq 1$.

 

[PeterDonis]

I assumed that's because the space is "contracted" radially

You should not be assuming anything. You should be learning how to do the math and understand the results yourself. Trying to make assumptions based on ordinary language statements when you don't understand the underlying math is not a good idea.

what the c in the Wikipedia's proper time equation would be if not the one measured in a local IFR?

You should not be trying to learn physics from Wikipedia. You need to learn it from a textbook. Sean Carroll's online lecture notes on GR would be a good place to start.

The short answer to the question you pose here is the one @PeroK gave: $c$ in GR is a conversion factor between units of length and units of time. Choosing $c = 1$ means choosing units in which length and time have the same units (e.g., seconds and light-seconds, or years and light-years, or nanoseconds and feet). It has nothing to do with anything anybody measures.

 

[Pyter]

You should not be trying to learn physics from Wikipedia. You need to learn it from a textbook.

I did study more than one book, but they didn't cover the topic of the invariance of velocity of light in GR. Even Einstein in his papers and divulgative book is rather reticent about that.

 

[PeterDonis]

I did study more than one book, but they didn't cover the topic of the invariance of velocity of light in GR.

Which books have you studied?

 

[Pyter]

@PeterDonis

• Intro to Differential Geometry and General Relativity - S. Waner
• Generalized Relativity - P. A. M. Dirac (which I sort of consider a "cram sheet", only 77 pages but very densely packed)
• Special Relativity and Classical Field Theory - Leonard Susskind et al. (SR only)

 

[PeterDonis]

Intro to Differential Geometry and General Relativity - S. Waner

This one doesn't discuss the invariant light cone structure of spacetime? That is what "invariance of velocity of light" translates to in GR.

The other two I can understand, IIRC Dirac doesn't discuss geometry much at all, and the Susskind one, since you say it focuses on SR, wouldn't be expected to discuss generalizations of concepts to GR. (Also it seems more focused on field theory.)

 

[Pyter]

I forgot to add: also the Einstein original papers and (partly) his "divulgation" book.
After all this, if it wasn't for your informative answers, I never could've guessed that the measured velocity of light could be > 1.

 

[PeroK]

After all this, if it wasn't for your informative answers, I never couldn't have guessed that the measured velocity of light could be > 1.

Take a look at this:

https://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html

 

[Pyter]

@PeroK

Einstein talked about the speed of light changing in his new theory. In the English translation of his 1920 book "Relativity: the special and general theory" he wrote: "according to the general theory of relativity, the law of the constancy of the velocity [—Either Einstein or his translator obviously mean "speed" here, since velocity (a vector) is not in keeping with the rest of his sentence. People often say "velocity" when they clearly mean "speed".] of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity [...] cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity [speed] of propagation of light varies with position." This difference in speeds is precisely that referred to above by ceiling and floor observers.

That's the very excerpt I cited earlier in this thread. And that's pretty much all Einstein says about it.

When all is said and done, to insist that a non-c speed of light is nothing more than an artifact of a "nonphysical" choice of coordinates is to make a wrong over-simplification.

Interesting.