Can gravitational time dilation explain the photon's deflection near the Sun?

[jeremyfiennes]

For a photon passing close to the Sun, Newtonian physics predicts a deflection of 0.85^o. GR gives the correct 1.7^o. Can the true value alternatively be obtained via 1) a Newtonian model, and 2) gravitational time dilation: the photon's slower speed near the Sun leads to it spending more time in its gravitational field, and hence a higher deflection?

 

[PeterDonis]

For a photon passing close to the Sun, Newtonian physics predicts a deflection of 0.85^o.

Actually, the "prediction" of Newtonian physics is not at all clear for this case, because it's not clear in Newtonian physics that light should be affected by gravity at all. The deflection you refer to is more like a prediction of the Newtonian approximation of GR, where we agree that light is affected by gravity but don't fully take into account all relativistic effects because the Newtonian approximation assumes that everything is moving much slower than light. And since that approximation clearly does not apply to light, this so-called "Newtonian prediction" is really based on an inconsistent model to begin with.

Can the true value alternatively be obtained via 1) a Newtonian model, and 2) gravitational time dilation: the photon's slower speed near the Sun leads to it spending more time in its gravitational field, and hence a higher deflection?

The true value is obtained by doing the relativistic case properly in the first place, recognizing that light cannot be treated by any "Newtonian" approximation. Various sources will tell you various stories in ordinary language about why the true result is what it is; some such stories will indeed include language like "the photon's slower speed near the Sun leads to it spending more time in its gravitational field". I would not recommend taking any such stories too seriously.

 

[Ibix]

Newtonian physics predicts a deflection of 0.85°. GR gives the correct 1.7°.

Incidentally, these figures are arcseconds, not degrees. In degrees the predicted deflections are about 0.00025° and 0.0005°.

 

[Dale]

it's not clear in Newtonian physics that light should be affected by gravity at all. The deflection you refer to is more like a prediction of the Newtonian approximation of GR, where we agree that light is affected by gravity but don't fully take into account all relativistic effects because the Newtonian approximation assumes that everything is moving much slower than light.

I disagree somewhat with this. I agree that it is unclear if light should be affected by gravity in Newtonian gravity. But I don’t think that making the choice that it is affected by gravity involves an approximation to any other theory.

Strictly within Newtonian gravity you can look at two possible formulations of the gravitational law:

$F=GMm/r^2$ or $a=GM/r^2$

The second formulation is clear that massless objects also are accelerated by gravity, while the first is unclear. The equivalence principle is also compatible with Newtonian gravity, and Newtonian gravity can be geometrized.

 

[jeremyfiennes]

Actually, the "prediction" of Newtonian physics is not at all clear for this case, because it's not clear in Newtonian physics that light should be affected by gravity at all. ... Some stories will indeed include language like "the photon's slower speed near the Sun leads to it spending more time in its gravitational field". I would not recommend taking any such stories too seriously.

So when I read, for instance in http://www.einstein-online.info/spotlights/light_deflection.html, that

"Theories of the deflection of light by mass date back at least to the late 18th century when the Reverend John Michell reasoned that were the Sun sufficiently massive light could not escape from its surface. Isaac Newton had noted in his 1704 Opticks that light particles should be affected by gravity. In 1801 the German astronomer Johann Georg von Soldner showed that rays from a distant star skimming the Sun's surface would be deflected through an angle of about 0.9 seconds of arc, based on Newton's laws and the assumption that light behaves like very fast moving particles."
​

this is basically all wrong?

 

[Nugatory]

this is basically all wrong?

Those statements were based on a then-contemporary assumption that light consisted of particles with small but non-zero mass. In fact, that assumption is stated explicitly in the last sentence you quote.

Perhaps the best way of stating things is "Classical physics says that light may or may not be deflected by gravity; if it is, the deflection will be half what general relativity predicts".

 

[jeremyfiennes]

I have been led to believe that photons have relativistic mass hν/c², which is what gravity acts on. The classical assumption was correct, even thought they didn't know the real reason why. The classical approach does not however include gravitational time dilation, then unknown. This leads back to my original question: can the correct result be obtained by including it.

 

[Vanadium 50]

I have been led to believe that photons have relativistic mass hν/c2, which is what gravity acts on

Then you have been led astray.

 

[Ibix]

I have been led to believe that photons have relativistic mass hν/c², which is what gravity acts on.

In relativity, gravity doesn't "act on" any particular aspect of a particle. Particles follow geodesics because that's the extremal path. And in Newtonian physics it's not at all clear what one means by "photon" since they are a product of relativistic quantum field theories.

The classical assumption was correct,

That seems rather doubtful. The basic problem with Newtonian gravity and corpuscular theories of light (even if, as @Dale proposes, you use acceleration not force) is that light speed varies. The 0.85" figure comes from considering a particle with a velocity at infinity of c, but a higher speed near the Sun. Totally fine in Newtonian physics, but totally inconsistent with Maxwell's equations.

Patching time dilation on top of that in the hope of rescuing something sounds like a fairly huge challenge. I very much doubt it's possible to do coherently - for example you ought to be able to set up a braking orbit around multiple planets that would eventually bring light to rest. In fact, it was effectively trying to patch something on top of Maxwell to make it consistent with Newtonian physics that led to the rejection of Newton in the first place.

 

[A.T.]

For a photon passing close to the Sun, Newtonian physics predicts a deflection of 0.85^o. GR gives the correct 1.7^o. Can the true value alternatively be obtained via 1) a Newtonian model, and 2) gravitational time dilation: the photon's slower speed near the Sun leads to it spending more time in its gravitational field, and hence a higher deflection?

The gravitational time dilation part is the local effect that replaces the Newtonian gravity. The additional bending comes from global spatial geometry.
http://demoweb.physics.ucla.edu/content/10-curved-spacetime
https://www.mathpages.com/rr/s8-09/8-09.htm

 

[jeremyfiennes]

Incidentally, these figures are arcseconds, not degrees. In degrees the predicted deflections are about 0.00025° and 0.0005°.

Ok. Sorry, I was (mis)quoting from memory.

 

[jeremyfiennes]

In Newtonian physics it's not at all clear what one means by "photon".

For present purposes a photon can be simply "whatever causes a displaced image of a star on a photographic plate".

 

[Dale]

Perhaps the best way of stating things is "Classical physics says that light may or may not be deflected by gravity; if it is, the deflection will be half what general relativity predicts".

I agree with this. It isn’t clear one way or the other that Newtonian gravity affects light, but if it does then the amount of deflection is unambiguous. It is either 0 or half the GR value. Either claim could be consistent with Newtonian gravity.

 

[jeremyfiennes]

The 0.85" figure comes from considering a particle with a velocity at infinity of c, but a higher speed near the Sun.

Wouldn't gravitational time diçation mean a slower speed near the Sun, leading to a greater deflection?

Patching time dilation on top of that in the hope of rescuing something sounds like a fairly huge challenge. I very much doubt it's possible to do coherently.

I am not proposing a patch. Valid physical results can be obtained via a number of routes. Gravitational time dilation is part of GR. I can't see how modifying the fixed-speed Newtonian calculation to include gravitational time dilation would be very complicated. But my maths is unfortunately not up to it. That is why I am asking you professionals.

 

[Ibix]

fixed-speed Newtonian calculation

It's not a fixed speed in Newtonian physics - that's the point I was making.

 

[jeremyfiennes]

It's not a fixed speed in Newtonian physics - that's the point I was making.

The speed of light was first calculated by Ole Römer in 1676. Soldner in 1801 presumably used his value.

 

[Ibix]

The speed of light was first calculated by Ole Römer in 1676. Soldner in 1801 presumably used his value.

He did use that as an initial value. I re-did his maths this morning. But once it interacts with Newtonian gravity its speed is not constant - Soldner was essentially modelling a ray of light as a stream of very high velocity machine gun bullets.

That's one reason why Newtonian gravity is inconsistent with relativity.

 

[jeremyfiennes]

He did use that as an initial value. I re-did his maths this morning. But once it interacts with Newtonian gravity its speed is not constant.

I was not aware of that. I assumed he had used a constant speed. But the acceleration on approaching the Sun will be offset by the deceleration when leaving it, so the net difference should not be much. I'm still interested in the effect gravitational time dilation would have, if also taken into account.

 

[A.T.]

I'm still interested in the effect gravitational time dilation would have, if also taken into account.

See post #10. Gravitational time dilation replaces Newtonian gravity, rather than being an addon.

 

[Ibix]

I was not aware of that. I assumed he had used a constant speed.

Why would he assume that? Einstein was inspired to think that way by the work of Maxwell and/or Lorentz, neither of whom was even born at the time Soldner was working. The speed gain from infall in the Newtonian model is tiny - about 640m/s on top of the original 300,000,000 - but that is fundamentally incompatible with relativity. And with a model of light as a wave.

Anyway, as A.T. reminds us, Newtonian gravity emerges from the time-time component of the Einstein field equations, dependent on the same part of the metric that gives us time dilation. You need to account for spatial curvature. But then you just have GR.

 

[pervect]

For a photon passing close to the Sun, Newtonian physics predicts a deflection of 0.85^o. GR gives the correct 1.7^o. Can the true value alternatively be obtained via 1) a Newtonian model, and 2) gravitational time dilation: the photon's slower speed near the Sun leads to it spending more time in its gravitational field, and hence a higher deflection?

The correct deflection value can be obtained by the PPN approximation, which is a weak-field approximation to General relativity. In this PPN formulation, there is exactly one parameter that controls the amount of the 'extra' deflection of the photon, this parameter is usually caled $\gamma$.

See for instance the wiki article on PPN, <<here>>.

The description of the PPN parameter $\gamma$ has nothing direclty due to gravitational time dilation, but is described in the wiki article (and in textbooks) as

$\gamma$ : How much space curvature $g_{ij}$ is produced by unit rest mass ?

The GR prediction has $\gamma=1$. Setting $\gamma=0$ cuts the deflection in half. Unfortunately, it's not quite correct to say that the Newtonaion prediction corresonds to setting $\gamma=0$.

It's clear, historically, that the "extra" deflection of light was considered to be a good test of General Relativity. WIki <<link>> has the following to say about it, but I haven't read the citied references.

Henry Cavendish in 1784 (in an unpublished manuscript) and Johann Georg von Soldner in 1801 (published in 1804) had pointed out that Newtonian gravity predicts that starlight will bend around a massive object.[15][16] The same value as Soldner's was calculated by Einstein in 1911 based on the equivalence principle alone. However, Einstein noted in 1915 in the process of completing general relativity, that his (and thus Soldner's) 1911 result is only half of the correct value. Einstein became the first to calculate the correct value for light bending.[17]

There are some other methods for getting the correct prediction of light for the deflection of light, such as using an action principle. But I don't think there's anything that's exactly what you describe in 2).

 

[jeremyfiennes]

The correct deflection value can be obtained by the PPN approximation

Ok. Thanks for a coherent answer that finally makes some sense to me. I was not aware of PNN. It seems to cover my query. In any situation, a valid model is anyone that gives a correct result, i.e. represents the experimental data. No single model has a monopoly. Since both the Newtonian and the GR models predict the Earth's orbit, both are valid in this particular case. It had seemed to me evident that a Newtonian model, incremented to account for factors not available to him (e.g. gravitational time dilation), should be able to at least approximate the correct result.
This seems about as far as I can go. So thank you again for your answer, and if all agree: 'thread closed'.

 

[Dale]

No single model has a monopoly. Since both the Newtonian and the GR models predict the Earth's orbit, both are valid in this particular case.

I 100% agree with this.

It had seemed to me evident that a Newtonian model, incremented to account for factors not available to him (e.g. gravitational time dilation), should be able to at least approximate the correct result.

First, then it would no longer be Newtonian.

Second, the tricky thing is to make sure that your new model is self consistent. Simply saying “time dilation plus Newtonian gravity” doesn’t make that combination of concepts self consistent. The PPN formulation starts with a mathematical framework that is self consistent without physical motivation. The terms are then given physical meaning by reference to other physically motivated theories that predict values for those parameters.

 

[jeremyfiennes]

First, then it would no longer be Newtonian.

Sure.

First, then it would no longer be Newtonian.

Second, the tricky thing is to make sure that your new model is self consistent.

Ok. But not all models can cover all situations. If a Newton+time-dilation model predicts the correct deflection in this particular case, then it is valid in this particular case. With no claim to any extended validity.

 

[Ibix]

Ok. But not all models can cover all situations. If a Newton+time-dilation model predicts the correct deflection in this particular case,

It doesn't.

 

[jeremyfiennes]

It doesn't.

Could you demonstrate this?

 

[A.T.]

Ok. But not all models can cover all situations. If a Newton+time-dilation model predicts the correct deflection in this particular case, then it is valid in this particular case. With no claim to any extended validity.

There is a big difference between "does not cover all situations" and "gives a similar result in one particular case, by accident".

 

[Ibix]

Could you demonstrate this?

As noted three times already, Newtonian physics emerges from the time component of GR. The PPN approximation @pervect refers to allows you to "turn on" spatial curvature (edit: and/or other effects not relevant to this scenario) on top of that, getting the correct GR result.

 

[jeremyfiennes]

As noted three times already, Newtonian physics emerges from the time component of GR.

Newtonian physics emerged long before GR.

 

[jeremyfiennes]

There is a big difference between "does not cover all situations" and "gives a similar result in one particular case, by accident".

I don't see how "by accident" comes into it. Heisenberg apparently stumbled onto his matrix notation "by accident" - i.e. no-one knows how he got to it. But it nevertheless works. And is therefore by definition valid.