Gravity AND curved space-time bending light

[bockerse]

Can anyone explain in words (or equations if you can't use words, or in words to describe the equations) how "...according to the theory [of gen relativity], half of this deflection [of light by any massive body, the sun in this quote's case] is produced by the Newtonian field of attraction of the sun and the other half by the geometrical modification ("curvature") of space caused by the sun"? The preceding quote [brackets excluded] is from a book entitled "Relativity" by Einstein himself, but he doesn't detail how HALF the bending is from space-time curvature and HALF from gravity. How can those contributions each be figured out, and are they each precisely half?

Thanks!
Gerrit

 

[Bill_K]

The wrong Newtonian calculation is given http://www.theory.caltech.edu/people/patricia/lclens.html

 

[pervect]

Can anyone explain in words (or equations if you can't use words, or in words to describe the equations) how "...according to the theory [of gen relativity], half of this deflection [of light by any massive body, the sun in this quote's case] is produced by the Newtonian field of attraction of the sun and the other half by the geometrical modification ("curvature") of space caused by the sun"? The preceding quote [brackets excluded] is from a book entitled "Relativity" by Einstein himself, but he doesn't detail how HALF the bending is from space-time curvature and HALF from gravity. How can those contributions each be figured out, and are they each precisely half?

I don't think there's a better way to explain it in words than the words you already have.

If you want an explanation that is not in words, you need to decide how exactly you are going to mathematically represent the curvature of space-time. There are several possible approaches that I'm aware of , and all of them give the same answer as your source does.

You can represent the curvature of space-time via Christoffel symbols, $$\Gamma^a{}_{bc}$$

These yield an equation known as the geodesic equation (expressed in terms of these Christoffel symbols and coordinate "velocities") that predicts how light propagates.

http://en.wikipedia.org/w/index.php?title=Solving_the_geodesic_equations&oldid=425732007

Using this approach you can separate the Christoffel symbols into those which give rise to the usual Newtonian "force", and others which represent the curvature of space, and show that the contribution of the first set and the second set are equal when computing the path light takes.

In a somewhat similar manner, you can represent the curvature of space-time via the Riemann curvature tensor, then you can use the Bel decomposition of this tensor and the geodesic deviation equation to show that half the geodesic deviation of light is from the terms that represent purely spatial curvature.

Finally, you can use the PPN formalism (which can be used to describe metric theories of gravity other than GR). http://en.wikipedia.org/w/index.php?title=Parameterized_post-Newtonian_formalism&oldid=441595158

In the PPN formalism, there's a single parameter, $\gamma$ which can be described in words as "the amount of space curvature produced by a unit rest mass.

In GR, the value of the PPN parameter $\gamma$ is 1, and using the PPN formalism to calculate light trajectories, you can show that you get double the deflection of light when$\gamma$ is equal to one than when it is zero.

You can also show that the deflection of light is insensitive to the various other PPN parameters (see the wiki page for a list of them), i.e. that $\gamma$ is the only PPN parameter that is important to light deflection.

So take your pick. Note that in all cases it's assumed that you're splitting space-time into space+time by considering "time" to be the time of a static observer.