First Experimental Confirmation of GR

[phoenix-anna]

TL;DR

Looking for an explanation of the curved path of light as it passes the sum

Trying to understand how Einstein predicted the appearance of stars during the solar eclipse around the turn of the 20th century (the first experimental confirmation of General Relativity, I believe). My impression, which is not physically correct, is that the sun attracts the photons as they pass by, causing their trajectory to bend. Actually, in GR there is no such thing as gravitational force, right? It is a matter of curvature. I understand that particles with mass follow geodesics, paths where the length (using the relativistic metric, I assume) is minimal. HOwever, I don't believe that GR treats photons as particles having mass but rather treats light as electromagnetic radiation. So that brings me to my question: what is relativity's explanation for the shift in the apparent position of these stars? What is the status of electromagnetic radiation in GR?

My confusion is amplified by my impression that in relativity, light always travels a light-like path where the length is zero: s^2=c^2 t^2-x^2=0

 

[Nugatory]

Photons only appear when we do a full quantum mechanical treatment of electromagnetic radiation, so are basically irrelevant to discussions of relativity; instead we treat light as classical electromagnetic radiation covered by Maxwell’s laws.

Light travels in a straight line, but in a curved spacetime the straight line paths don’t behave the way they do in Euclidean space: initially parallel lines may converge or diverge and end up heading in different directions. That’s why light appears to be deflected as it passes through the curved spacetime near a massive body.

My confusion is amplified by my impression that in relativity, light always travels a light-like path where the length is zero: s^2=c^2 t^2-x^2=0

That is true only in flat spacetime. In curved spacetime the length (better called the “interval” because it’s not the length of anything) of a lightlike path is still zero, but the equation describing it is appreciably more complicated and ends up describing a curved path (the solution to your equation with $s^2=0$ is the straight line path $x=ct$). To find the appropriate equation for the path of light in a curved spacetime we have to solve the “geodesic equation”; Einstein did that for the spacetime around the sun to calculate the expected path of light around the sun.

 

[PeterDonis]

What is the status of electromagnetic radiation in GR?

As @Nugatory said, GR is a classical theory so we treat light using Maxwell's Equations, adapted for use in curved spacetime.

For the case of light bending by the Sun, we can further simplify things by using what is called the geometric optics approximation, where we treat light as rays following null geodesic worldlines in curved spacetime. That approximation is what @Nugatory was describing in his post.

 

[Dale]

Summary:: Looking for an explanation of the curved path of light as it passes the sum

I understand that particles with mass follow geodesics, paths where the length (using the relativistic metric, I assume) is minimal. HOwever, I don't believe that GR treats photons as particles having mass

That is correct. Light also travels on geodesics, but they are not the same geodesics in which massive objects travel. They are null geodesics. The concept is similar, but with some slight differences. A null geodesic extremizes the path length, but actual interval will always be zero. Also, since proper time is undefined on a null path you need to use an affine parameter instead. Those two issues are more or less just technical mathematical subtleties, and conceptually it is the same general idea as for massive particles.

 

[PAllen]

In addition to what others have said, from a given point in spacetime (event), in either SR or GR, there are geodesics (locally straight paths) going in all directions that can be parameterized by 3 parameters (two spatial angles, and a hyperbolic angle called rapidity, which represents an "angle in the timelike direction"). Two light cones are defined by any choice of spatial angles, and a particular rapidity that produces null interval along the path - the path followed by light pulses. These null geodesics define light cones. The interior of these contain the timelike geodesics, the exterior contain the spacelike geodesics.

 

[PeterDonis]

a particular rapidity that produces null interval along the path

Actually, the "rapidity" corresponding to a null interval would be infinite, yes?

 

[A.T.]

So that brings me to my question: what is relativity's explanation for the shift in the apparent position of these stars? What is the status of electromagnetic radiation in GR?

This is a good description:
https://www.mathpages.com/rr/s8-09/8-09.htm

 

[PAllen]

Actually, the "rapidity" corresponding to a null interval would be infinite, yes?

Right, and for spacelike directions, the rapidity becomes a complex number. I should have used a different third parameter, perhaps simply $\beta$, so lightlike =1, timelike < 1, spacelike > 1, to avoid such complications.

 

[vanhees71]

You can just use an arbitrary parameter to describe a null geodesic. Most convenient are affine parameters of course. You just extremize the action
$$S[q]=\int \mathrm{d} \lambda \frac{1}{2} g_{\mu \nu} \dot{q}^{\mu} \dot{q}^{\nu}.$$
This "square form" has the great advantage that you can use it also in the limit of "massless particles" (though photons are not massless particles as already stressed above). The fact that the Lagrangian doesn't explicitly depend on $\lambda$ implies that the "Hamiltonian"
$$H=\dot{q}^{\mu} \frac{\partial L}{\partial \dot{q}^{\mu}}-L=L=\text{const}$$
along the solutions of the equations of motion.

For "massless particles" $L=0$, i.e., you solve the equations of motion with $L=0$ to get the null geodesics of the spacetime. Then $\lambda$ has no specific physical meaning. It's just an arbitrary parameter parametrizing the null geodesics.

For massive particles you have $L>0$ (west-coast convention) and you can choose $\lambda=\tau$, the particle's proper time along it's path by setting $L=c^2/2$.