General relativity and light

[Ring]

Why are the paths of two photons moving in opposite directions bent by double the amount that a pair of massive particles would experience?

Does a single photon passing a gravitating body experience double the deflection a massive particle would?

I would like to thank you in advance for your expected outstanding answers.

[JS_Princeton]

I didn’t want to be the first one to reply… because I’m just now taking GR and don’t understand it all completely myself. There really is no simple explanation I can point to to why this ends up happening this way other than to say that the solutions to the Field Equations work out that way. There are definitely books you can pick up on the subject. For a lay intro to GR, I recommend my professor’s book Time Travel in Einstein’s Universe (R. Gott, Houghton Mifflin, 2001). Otherwise, some serious tensor and metric learning is in order.

Maybe you’ve done that, in which case we can go through the equations and see how a photon ends up deflected by twice the amount predicted by Newton when travelling in a vacuum near a mass. (by the by, one of the interesting results of the field tensor is that gravity is not only produced by mass, but also by pressure!)

[Ring]

I found the following:

http://casswww.ucsd.edu/public/tutorial/GR.html

1.Deflection of Light by Gravity: A direct consequence of the equivalence principle is that light should be deflected or bent by gravity. Einstein twice calculated the amount that light would be deflected passing by the sun, the largest “nearby” mass. His first calculation used only the Equivalence Principle and the equivalent mass-energy of a visible photon. In his second calculation, published in 1916, he included the space-time metric, which describes the curvature of space and time caused by gravity and got an answer twice as large as his first calculation. The second calculation predicts that light from a distant star passing by the limb of the sun would be deflected by 1.75 arcseconds (less than 1/2000th of a degree).

This pretty much explains why the bending of light is twice what Newton would have predicted, but it doesn’t answer why the bending of light is twice what it would be for a particle with a rest mass.

[ZenBeam]

Ring, are you sure a massive particle does have half the deflection of a photon? In Weinberg’s Gravitation and Cosmology, p 19, he writes

The Principle of Equivalence of Gravitation and Inertia […] determines the effects of gravitation on arbitrary physical systems, but it does not determine the field equations for gravitation itself. Einstein tried to use the equivalence principle in 1911 to calculate the deflection of light in the sun’s gravitational field, but the structure of the field was not then correctly understood and Einstein’s answer was one-half the “correct” general-relativistic result.

It sounds like the massive particle calculation isn’t GR, but rather a first crack at gravitation satisfying the principle of equivalence.

[Ring]

Zenbeam wrote:

Ring, are you sure a massive particle does have half the deflection of a photon? In Weinberg’s Gravitation and Cosmology, p 19, h

No I’m not sure, but the following was posted in sci.physics and no one disagreed. And I have yet to see someone post something there and get away with a false statement.

There is one significant difference between a photon and a massive particle.
All massive particles (particles with rest mass) are attracted to each other
(in the Newtonian limit) unidirectionally with the square of the distance
between them. Light, on the other hand, (in GR only) is attracted by twice
that amount to mass. But then, light’s attraction is directionally limited.
If two beams of light are propagating in parallel, they will not attract at
all. However, if they are antiparallel, they attract more than they would if
both were simply equivalent rods of mass

If this really intrigues you, get R.C. Tolman’s classic book “Relativity,
Thermodynamics and Cosmology” first published in 1934, now available from
Dover rather reasonably priced. See Ch 8, Relativistic Electrodynamics, Part
II, section 112 "The Gravitational Field Corresponding to a Directed Flow of
Radiation. also 113 and 114 for the gravitational field of a pencil of
light, and then a pulse of light.

[Chronos]

A single photon will also be deflected by twice the Newtonian amount in passing by a massive object-- You don’t need a pair of photons. Does this provide an easy way to distinguish between a massless particle and a massive particle? Not really… A very lightweight particle, like a neutrino (which is hence moving at very close to the speed of light, at a given energy) will be deflected by an amount almost equal to the amount for light. There is no experimental evidence, nor is there likely ever to be, that the photon is truely massless; all we can say experimentally is that if the photon does have mass, it’s an insanely small one.

[ZenBeam]

A single photon will also be deflected by twice the Newtonian amount in passing by a massive object

Chronos, my reading of the OP, and the quote from Ring’s subsequent post, is that the question isn’t whether the photon is deflected by twice the Newtonian amount. It’s whether a photon, under GR, is deflected by twice the amount of a massive particle, also under GR.

Are you saying the answer to “Does a single photon passing a gravitating body experience double the deflection a massive particle would?” is “no”? And, of course, do you have a cite or derivation? Certainly “yes” would be a counter-intuitive answer (to me, anyway), but Ring’s quote seems to say that “yes” is correct.

[ZenBeam]

Surely someone here can answer this…

[bonzer]

I’m not sure Chronos’ point was that unclear: a nearly massless particle will have a deflection in passing a massive body that will be almost indistinguishable from that for a truely massless one. The latter happens to be twice the Newtonian result, but this is no more than a convenient comparison - it is, after all, wrong.

More formally, Chronos is claiming that the deflection has a well-behaved limit in the limit of the mass tending to zero and that this is equal to the massless case. I don’t have an explicit cite - and I doubt Chronos can dig one up either - but that’s largely because physicists take such statements as obvious, unless there’s some trap. As you say, ZenBeam, it would be counter-intuitive. And rather than give a derivation, I’m going to point in the direction of section 11.1 in Schutz A First Course in General Relativity (Cambridge, 1985), where he lays out the orbit equations in the massive and massless cases alongside each other. The key comparison is between (11.8) and (11.9), which superficially differ by one extra term in the massive case. However, if you put the mass of the particle in explicitly, then it’s clear that that term vanishes in the m tends to zero limit. Not entirely rigorous, but you’re always free to slog through the full derivations of the deflections in both cases.

Frankly, I don’t understand the quote Ring passed on and, unfortunately, I don’t have immediate access to Tolman’s book.

Incidently, I don’t know of any intuitive argument for why the massless answer comes out exactly twice the Newtonian answer. I seem to remember that Weinberg gives a slick derivation of Rutherford scattering in an appendix to The Discovery of Subatomic Particles (?), up to an undetermined overall constant. I suspect you could similarly argue that the form of the deflection formula is fixed by such an argument and so the only thing that can differ is a simple numerical factor. But why exactly 2?

[Ring]

There is one significant difference between a photon and a massive particle. All massive particles (particles with rest mass) are attracted to each other in the Newtonian limit) unidirectionally with the square of the distance
between them. Light, on the other hand, (in GR only) is attracted by twice that amount to mass.

The underlined portion is the question. Does this have anything to with lightlike geodesics versus timelike geodesics?

[ZenBeam]

I think I have part of it. First, some equations from Pauli’s Theory of Relativity (slightly rearranged):

For a point mass:
1 / r[sup]4[/sup] * (dr / d[sym]f[/sym])[sup]2[/sup] + 1/r[sup]2[/sup] - 2 m c[sup]2[/sup] / (B[sup]2[/sup]r) - 2m/r[sup]3[/sup] = 2 E / B[sup]2[/sup] (Eq 427 a)

For a photon:
1 / r[sup]4[/sup] * (dr / d[sym]f[/sym])[sup]2[/sup] + 1/r[sup]2[/sup] - 2m/r[sup]3[/sup] = 1 / [sym]D[/sym][sup]2[/sup] (Eq 427 b)

where (he writes) “the law of areas
r[sup]2[/sup] * d[sym]f[/sym] / d[sym]t[/sym] = const. = B (Eq 425)
is seen to be valid”

Note that m is not the particle mass, but rather is proportional to the mass generating the field (e.g. the Sun, not the proton zipping by). Also not that the variable in Eq. (425) is proper time [sym]t[/sym], not observer time t. [sym]f[/sym] and r are normal polar coordinates.

He writes “These equations completely determine the required paths”. “If the last term on the left-hand side [of equation (427 b)] were not present, the light ray would be a straight line, at a distance [sym]D[/sym] from the origin.”

At any rate, as the speed of the massive particle increases towards C, d[sym]f[/sym] / d[sym]t[/sym] at the minimum distance from the Sun increases without bound. This means the term 2 m c[sup]2[/sup] / (B[sup]2[/sup]r) in (427 a) approaches zero. He doesn’t say what E is, but energy seems a good guess. In that case, E will also increase without bound, presumably keeping E / B[sup]2[/sup] finite.

All that’s left is to show 1 / [sym]D[/sym][sup]2[/sup] = 2 E / B[sup]2[/sup] in the limit as the massive particle speed approaches C (or not, as the case may be).

Love these [sym]symbol fonts[/sym]