GR. Einstein's 1911 Paper: On the Influence of Gravitation on the Propagation of Light

Regarding the paper, what does Einstein means when he says:

"If we call the velocity of light at the origin of co-ordinates $c_0$, then the velocity of light $c$ at a location with the gravitation potential $\Phi$ will be given by the relation: $c = c_0\cdot\left(1+\frac{\Phi}{c^2}\right).$ The principle of the constancy of the velocity of light holds good according to this theory in a different form from that which usually underlies the ordinary theory of" relativity"

Does the velocity of light is constant only into a space where the gravitation potential is constant either?

• The coordinate velocity of light is only constant in regions of no potential. In practice however note that usually $|\Phi| \ll 1$ except for special cases like black holes ($=1/2$) and neutron stars so the relative change is not very big. – Kibble Dec 8 '16 at 21:56
• (Maybe off topic) May I suggest studying GR using modern literature? Einstein was obviously a genious and the person behind GR originally, but his papers just aren't well-suited for beginners for studying the subject. Nowadays GR has become renowned and a great deal of literature for all levels (introductory to professional) has emerged. – Prof. Legolasov Dec 9 '16 at 0:37

There is a lot of confusion about what exactly is meant by the speed of light in general relativity, so I think it’s worth examining this with some care. The issue turns out to be absolutely fundamental to general relativity.

Special relativity

Let’s start with special relativity. Although it’s rarely introduced as such, special relativity is simply the flat spacetime limit of general relativity i.e. it is the spacetime geometry described by the Minkowski metric:

$$\mathrm ds^2 = -~c^2~\mathrm dt^2 + \mathrm dx^2 + \mathrm dy^2 + \mathrm dz^2 \tag{1}$$

where we are using the time coordinate $t$ and the Cartesian spatial coordinates $x$, $y$ and $z$. The parameter $c$ is a constant, and for now let’s not make any assumptions about it, though we’ll see that it turns out to be the speed of light.

Light (and any massless particles) travel on null trajectories i.e. those trajectories where $\mathrm ds = 0$, and if we substitute this into equation (1) we get after some minor rearranging:

$$c = \frac{\sqrt{ \mathrm dx^2 + \mathrm dy^2 + \mathrm dz^2}}{\mathrm dt}$$

But $\sqrt{ \mathrm dx^2 +\mathrm dy^2 +\mathrm dz^2}$ is just the Pythagoras’ expression for the total distance moved in space - let’s call this $\mathrm dr$ - so we get:

$$c = \frac{\mathrm dr}{\mathrm dt}$$

And this is just the velocity of the light so:

the constant $c$ is the velocity of light

The Minkowski metric is unchanged by any Lorentz transformation, so all observers related by Lorentz transformations will measure the speed of light to have the same constant value $c$. This is what we mean when we say that in special relativity the speed of light is constant.

But even in special relativity things are not as simple as they initially appear. The metric (1) describes the spacetime for an inertial observer, and the Lorentz transformations relate the frames of inertial observers. However it’s possible to have accelerated observers, for example observers in a rocket accelerating with some acceleration $a$, and to get the metric for an accelerated observer we need to use a Rindler transformation. This gives us the Rindler metric for a proper acceleration $a$:

$$\mathrm ds^2 = -~ \left(1 + \frac{a}{c^2}x \right)^2 c^2~\mathrm dt^2 +\mathrm dx^2 + \mathrm dy^2 + \mathrm dz^2 \tag{2}$$

If we use the same trick as before to calculate the speed of light we get:

$$\frac{\mathrm dr}{\mathrm dt} = c \left(1 + \frac{a}{c^2}x \right) \tag{3}$$

And we discover that the speed of light is not constant but varies by a factor of $\left(1 + \frac{a}{c^2}x \right)$ where $x$ is the distance from the observer. What’s going on? Well, there are two key points to make

Firstly, this is the same light that the non-accelerating observer measures to have the speed $c$, and the light hasn’t changed. What has changed is that the accelerating observer’s spatial coordinates are curved due to the acceleration. The different speed of light is due to a change in the coordinates not a change in the light. For this reason we refer to the speed we calculated above as the coordinate velocity of light, that is it’s the velocity measured in whatever coordinates the observer is using. When those coordinates are curved the velocity will generally be different from $c$.

Secondly, even though the speed of light can vary in the accelerating frame, let’s see what happens at the position of the observer i.e. at $x = 0$. When we substitute $x=0$ equation (3) simplifies to:

$$\frac{\mathrm dr}{\mathrm dt} = c$$

And we find the speed of light at the observer’s position is $c$, just like the non-accelerating observer. We call this the local velocity of light because it’s measured locally i.e. at the observer’s position.

So what we’ve found is:

1. The coordinate velocity of light can be different from $c$

2. The local velocity of light is still $c$

General relativity

Now on to general relativity. In general relativity we describe spacetime as a Lorentzian manifold, and solving Einstein’s equation gives us the shape of this manifold i.e. the metric. To write down the metric we need to choose a coordinate system, and in GR all coordinate systems are equally valid and we can choose whatever coordinate system we want. Generally we try to choose coordinates that make our calculations easiest.

For our purposes the key point is that for a Lorentzian manifold there is always a choice of coordinates that makes the spacetime geometry locally the Minkowski metric i.e. there is always a choice of coordinates that makes spacetime locally flat. These coordinates correspond to the rest frame of a freely falling observer, so for a freely falling observer it is as if they are at rest in a flat spacetime. This is only true locally, and as we move away from the freely falling observer the spacetime curvature will cause tidal forces, however at the position of the observer the tidal forces go to zero.

But we already know from the discussion above that in a Minkowski spacetime the speed of light is the constant $c$, and that means our freely falling observer always measures the local speed of light to be the constant $c$.

But what of observers who aren’t falling freely? I’m going to gloss over this and just say that for any observer who isn’t freely falling the spacetime looks locally like a Rindler spacetime. And as we discussed above in a Rindler spacetime the local speed of light is also just the constant $c$. So even an accelerating observer also measures the speed of light to be the constant $c$.

So just like in special relativity we end with the conclusion:

The local velocity of light is $c$

And this is what we mean when we say the speed of light is constant in general relativity.

As we found earlier for accelerating observers in flat spacetime the coordinate velocity of light may not be equal to $c$. We’ll take the well worn example of the Schwarzschild metric that describes the spacetime geometry round a static black hole:

$$\mathrm ds^2 = -~\left(1-\frac{r_s}{r}\right)c^2~\mathrm dt^2 + \frac{\mathrm dr^2}{1 - r_s/r} + r^2~\left(\mathrm d\theta^2 + \sin^2\theta~\mathrm d\phi^2\right)$$

We calculate the coordinate speed just as before. We’ll assume the light ray is moving in a radial direction so that $\mathrm d\theta = \mathrm d\phi = 0$, and as before substituting $\mathrm ds=0$ gives us the coordinate speed:

$$\frac{\mathrm dr}{\mathrm dt} = c \left(1-\frac{r_s}{r}\right)$$

And we find the coordinate speed is not $c$. However the Schwarzschild coordinates are the coordinates for an observer at infinity, i.e. at $r=\infty$, so we get the local speed of light for this observer by substituting $r=\infty$ to get:

$$\frac{dr}{dt} = c \left(1-\frac{r_s}{\infty}\right) = c$$

So once again the local speed of light is $c$. I won’t do it since the algebra is a bit tedious, but it’s easy to show that for any observer in a Schwarzschild spacetime the local speed of light is also just the constant $c$. So as before we end up with the conclusion that for all observers in Schwarzschild spacetime the local speed of light is $c$.

And finally

I’ve diverged a long way from your original question, but my aim is to make the point that the local speed of light remains $c$ even in GR. The coordinate speed of light may not be $c$, but we can choose any coordinates we want and indeed different observers will in general be using different coordinates, so there is nothing special or unique about the coordinate velocity.

Having made this point let’s return to your question about the gravitational potential. Apart from a few special cases it is exceedingly hard to solve Einstein’s equation to get the metric. However if the gravitational fields are weak then we can use an approximation called the weak field limit. In this case the metric is:

$$\mathrm ds^2 \approx -\left( 1 + \frac{2\Delta\phi}{c^2}\right) c^2~\mathrm dt^2 + \frac{1}{1 + 2\Delta \phi/c^2}\left(\mathrm dx^2 + \mathrm dy^2 + \mathrm dz^2\right)$$

where $\Delta \phi$ is the difference in the Newtonian gravitational potential from the position of the observer. And calculating the coordinate speed of light as before gives us:

$$\frac{\mathrm dr}{\mathrm dt} = c \left( 1 + \frac{2\Delta \phi}{c^2}\right)$$

To get the local speed of light we note that at the observer’s position $\Delta\phi=0$ so the local speed of light is:

$$\frac{\mathrm dr}{\mathrm dt} = c$$

Surprise, surprise!

• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Dec 13 '16 at 5:13

There is a subtlety here. The impression is that in 1911 Einstein confirmed Newton's prediction (the speed of light falling towards the source of gravity varies like the speed of ordinary falling bodies), and in the final version of general relativity the variation was doubled. This is a wrong impression (Einstein did not predict that) but Einsteinians reinforce it from time to time:

http://arxiv.org/pdf/gr-qc/9909014v1.pdf Steve Carlip: "It is well known that the deflection of light is twice that predicted by Newtonian theory; in this sense, at least, light falls with twice the acceleration of ordinary "slow" matter."

What general relativity does predict is that the speed of falling light DECREASES (in the gravitational field of the Earth the acceleration of falling photons is NEGATIVE, -2g):

Quote: "Contrary to intuition, the speed of light (properly defined) decreases as the black hole is approached."

Quote: "Einstein wrote this paper in 1911 in German. (...) ...you will find in section 3 of that paper Einstein's derivation of the variable speed of light in a gravitational potential, eqn (3). The result is: c'=c0(1+φ/c^2) where φ is the gravitational potential relative to the point where the speed of light c0 is measured. Simply put: Light appears to travel slower in stronger gravitational fields (near bigger mass). (...) You can find a more sophisticated derivation later by Einstein (1955) from the full theory of general relativity in the weak field approximation. (...) Namely the 1955 approximation shows a variation in km/sec twice as much as first predicted in 1911."

Quote: "Specifically, Einstein wrote in 1911 that the speed of light at a place with the gravitational potential φ would be c(1+φ/c^2), where c is the nominal speed of light in the absence of gravity. In geometrical units we define c=1, so Einstein's 1911 formula can be written simply as c'=1+φ. However, this formula for the speed of light (not to mention this whole approach to gravity) turned out to be incorrect, as Einstein realized during the years leading up to 1915 and the completion of the general theory. (...) ...we have c_r =1+2φ, which corresponds to Einstein's 1911 equation, except that we have a factor of 2 instead of 1 on the potential term."

• The quotes here are completely useless because you did not define what definition for "speed of light" you are talking about. As JohnRennie's answer shows, the local speed of light is constant for all observers, the coordinate speed is not. The answer does nothing to clear up this confusion. – ACuriousMind Dec 9 '16 at 15:52

Regarding the paper, what does Einstein means when he says: "If we call the velocity of light at the origin of co-ordinates $c_0$, then the velocity of light $c$ at a location with the gravitation potential Φ will be given by the relation: $c = c_0*(1+(Φ/(c^2))).$ The principle of the constancy of the velocity of light holds good according to this theory in a different form from that which usually underlies the ordinary theory of relativity"

He meant exactly what he said. The speed of light varies with gravitational potential. It varies with altitude, with elevation. He said this repeatedly, year after year:

1912: "On the other hand I am of the view that the principle of the constancy of the velocity of light can be maintained only insofar as one restricts oneself to spatio-temporal regions of constant gravitational potential".

1913: "I arrived at the result that the velocity of light is not to be regarded as independent of the gravitational potential. Thus the principle of the constancy of the velocity of light is incompatible with the equivalence hypothesis".

1914: “In the case where we drop the postulate of the constancy of the velocity of light, there exist, a priori, no privileged coordinate systems".

1915: "the writer of these lines is of the opinion that the theory of relativity is still in need of generalization, in the sense that the principle of the constancy of the velocity of light is to be abandoned".

1916: “In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity”.

1920: “Second, this consequence shows that the law of the constancy of the speed of light no longer holds, according to the general theory of relativity, in spaces that have gravitational fields. As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable”.

Does the velocity of light is constant only into a space where the gravitation potential is constant either?

Yes. Note that it wasn't only Einstein who said this. See the Shapiro delay: "according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path". Also see Is The Speed of Light Everywhere the Same? by relativist Don Koks on the Baez/PhysicsFAQ website: "In that sense, we could say that the "ceiling" speed of light in the presence of gravity is higher than the "floor" speed of light". Of course, what's more important that what Einstein or anybody else said, is the hard scientific evidence. If you open up a clock, you don't see time flowing through it. Instead you see cogs turning, or a pendulum swinging, or a crystal oscillating, or something else moving. When that thing moves slower, the clock goes slower. Optical clocks follow the same principal, and they go slower when they're lower. See the David Wineland interview: "if one clock in one lab is 30 centimeters higher than the clock in the other lab, we can see the difference in the rates they run at". An optical clock is not some cosmic gas meter with time flowing through it. It goes slower when it's lower because light goes slower when it's lower, not for any other reason.

Note that there are people who will tell you the speed of light is absolutely constant. That's a popscience myth I'm afraid. Sadly it's sometimes accompanied by a great deal of mathematical handwaving and abstraction intended to impress the innocent. There are also people who will tell you that the local speed of light is always c. This is tautological nonsense I'm afraid. See https://arxiv.org/abs/0705.4507 where Magueijo and Moffat refer to it. We use the local motion of light to define the second as the duration of 9,192,631,770 periods of a particular radiation. The we define the metre as the distance travelled by light in vacuum in 1/299792458th of a second. When you then use the second and the metre to measure the local motion of light, you will "measure" the local speed of light to be 299,792,458 m/s by definition. Repeat this entire exercise at two different elevations, and you will measure the upper light pulse to be moving at 299,792,458 m/s, and you will also measure the lower light pulse to be moving at 299,792,458 m/s:

The two light pulses are moving at different speeds because the seconds at the two elevations are not the same.