I Bending of light - caused by gravity or relativity?

[Prometeus]

The mainstream interpretation of GRT equations is, that additional double amount of angle of bending of light (Newton vs GRT) is caused by gravity (which is interpreted as curvature of spacetime). But when looking on the equations, it seems that this additional amount of bending is caused by relativistic effect based on speed of the particle and not by gravity itself.

Examples explaning what I mean:
Newton prediction of bending angle for light is X, GRT prediction of bending angle of light is 2X
but let's take an massive particle, for example neutrino
At speed of let's say 1000 m/s around Sun, is the bending angle of neutrino 99,9999% of X (practically the same as Newton prediction), but when the neutrino has 99,999999999% of speed of light, then the bending angle would be around 99,9999999% of 2X, practically the same as angle for photon (which does not have rest mass).

So it seems to me that this additional bending of light is solely depending on relativistic effects caused by speed of particle and not by curvature of spacetime (gravity). I would be interested in some explanation, what is wrong with my understanding.

 

[Vitro]

So it seems to me that this additional bending of light is solely depending on relativistic effects caused by speed of particle and not by curvature of spacetime (gravity). I would be interested in some explanation, what is wrong with my understanding.

You're thinking about it backwards. There's no additional bending in GR but not enough bending in the Newtonian model as the speed of the particle is increased. What this should tell you is that you are straying away from conditions where the Newtonian gravitation model is a good enough approximation of the world, and the farther you stray the less accurate the approximation becomes. Newtonian gravitation works well for everyday practical purposes in low curvature and low speed situations, but you must remain aware of where the boundary of applicability is.

In GR there's actually no bending of a free-fall particle's path, they all travel on geodesics which are the equivalent of "straight lines" in a curved spacetime. The bending you perceive is a consequence of an arbitrary choice of coordinates.

 

[Prometeus]

You're thinking about it backwards. There's no additional bending in GR but not enough bending in the Newtonian model as the speed of the particle is increased. What this should tell you is that you are straying away from conditions where the Newtonian gravitation model is a good enough approximation of the world, and the farther you stray the less accurate the approximation becomes. Newtonian gravitation works well for everyday practical purposes in low curvature and low speed situations, but you must remain aware of where the boundary of applicability is.

In GR there's actually no bending of a free-fall particle, they all travel on geodesics which are the equivalent of "straight lines" in a curved spacetime. The bending you perceive is a consequence of an arbitrary choice of coordinates.

It seems that you missed the main point. Just ignore Newton. In GRT the gravity of Sun (supposed curvature) is the same all the time but the bending is mainly depending on the speed of the very same particle (neutrino) and not on the (supposed) curvature.

 

[Vitro]

It seems that you missed the main point. Just ignore Newton. In GRT the gravity of Sun (supposed curvature) is the same all the time but the bending is mainly depending on the speed of the very same particle (neutrino) and not on the (supposed) curvature.

Weren't you comparing the difference in deflection angle between Newtonian gravity and GR? You called one X and the other 2X (and presumably you think of X as gravitational bending), then called the difference from X to 2X "additional bending" which is not caused by gravity. What I'm telling you is that the 2X predicted by GR is all caused by gravity (curvature) and the Newtonian X is simply wrong. Newtonian mechanics and gravity doesn't work at relativistic speeds (or in very strong gravity).

The bending of the path depends on the speed of the particle in both theories, and at low speeds and low gravity their predictions are very similar as you know. So I don't get your objection.

 

[A.T.]

The mainstream interpretation of GRT equations is, that additional double amount of angle of bending of light (Newton vs GRT) is caused by gravity (which is interpreted as curvature of spacetime).

Historically, the curvature of a space-time slice (radial space coordinate + time) was matching the effect of Newtonian gravity. The "additional" half of light bending was added later when the curvature of space (radial space coordinate + tangential space coordinate) was considered.

http://www.mathpages.com/rr/s8-09/8-09.htm

 

[PeterDonis]

Historically, the curvature of a space-time slice (radial space coordinate + time) was matching the effect of Newtonian gravity.

I don't think this is correct. Newtonian gravity has no space curvature. When we calculate a Newtonian prediction for the bending of light by the Sun, we are modeling "light" as "a test particle moving at the speed of light". We are not modeling light the way modern, i.e., relativistic, physics models it, as "an entity that, in the classical approximation, moves on null worldlines", because there are no null worldlines in Newtonian physics.

The "additional" half of light bending was added later when the curvature of space (radial space coordinate + tangential space coordinate) was considered.

This isn't really correct either. The GR calculation, if you actually look at the math, does not have two separate pieces, one due to "curvature of space" and the other due to the "Newtonian" calculation (or one due to "radial space coordinate" and one due to "tangential space coordinate"). It just has one piece, which gives an answer twice as large as the "Newtonian" answer.

But even the "Newtonian" answer, here, is not calculated using Newtonian physics. It's calculated, as the web page you linked to notes, using the equivalence principle, which applies special relativity in a local inertial frame on a small patch of spacetime. By the EP, light, in a single local inertial frame, should "accelerate" towards the Sun by the same amount as any other test object. If we take that calculation and extrapolate it to infinity in both directions, to get a total deflection, the answer comes out half as large as the GR calculation I referred to above.

Various texts give various heuristic ways of explaining why this happens, of which "space curvature" is one. What all of these really amount to is that, since the spacetime around the Sun is curved in the GR model, all of the different local inertial frames along the trajectory of a light ray grazing the Sun do not "fit together" the way the EP-based calculation, when extrapolated as I just describes, assumes they do. The GR calculation does the "fitting together" correctly. The various heuristics are really just taking different viewpoints on how to describe the correct "fitting together". But that's a matter of words, not physics.

 

[A.T.]

The GR calculation, if you actually look at the math, does not have two separate pieces

That's why I wrote "historically". Initially Einstein considered only the radial direction + time. Adding the other spatial directions doubled the deflection.

 

[pervect]

The mainstream interpretation of GRT equations is, that additional double amount of angle of bending of light (Newton vs GRT) is caused by gravity (which is interpreted as curvature of spacetime). But when looking on the equations, it seems that this additional amount of bending is caused by relativistic effect based on speed of the particle and not by gravity itself.

Examples explaning what I mean:
Newton prediction of bending angle for light is X, GRT prediction of bending angle of light is 2X
but let's take an massive particle, for example neutrino
At speed of let's say 1000 m/s around Sun, is the bending angle of neutrino 99,9999% of X (practically the same as Newton prediction), but when the neutrino has 99,999999999% of speed of light, then the bending angle would be around 99,9999999% of 2X, practically the same as angle for photon (which does not have rest mass).

So it seems to me that this additional bending of light is solely depending on relativistic effects caused by speed of particle and not by curvature of spacetime (gravity). I would be interested in some explanation, what is wrong with my understanding.

I'm not sure what you mean by "relativistic effects". How would you tell if an effect was "relativistic" or "not relativistic"? If we assume that Newton's theory is "not relativistic" (that seems logical to me), then any deviation from Newtonian predictions would be due to "relativistic effects". And that extra bending is one of the predictions of GR, so the extra bending of light being due to "relativistic effects" becomes essentially a tautology.

We can say a bit more about the extra bending of light if we make more assumptions. The key assumption we make is that we have a metric theory of gravity. Without gravity, special relativity has a mathematical entity called the "Miknowski metric". A metric gravity of theory says that we still have the same basic metric structure of special relativity (where we started), when we go to the more complete theory of General Relativity which (unlike Special relativity) is able to handle gravity. The more general metric of GR won't be the "Minkowski metric", though.

When we assume we have a metric theory of gravity, we can use the results of what's called PPN theory. PPN theory will handle _any_ metric theory of gravity, not just General Relativity, though it is limited to weak fields (linearized gravity) by it's construction.

This is a lot of technical detail, going by pretty fast, but I'll give a link to PPN theory, <<link>>, and you can ask questions about it or about the more general topic of why we focus on so-called "metric theories of gravity". I believe though that saying that special relativity has a metric structure and that that leads us to think that General Relativity should also have such a structure is sufficient for an overview.

When one does a formal calculation from PPN theory, the bending of light depends only on the PPN parameter $\gamma$. This parameter is described as "How much space curvature $g_{ij}$ is produced by unit rest mass ?". Note that this is purely spatial curvature, not the more general space-time curvature.

Thus it seems reasonable to conclude that the "extra" bending of light can be ascribed to the curvature of space, more formally the PPN parameter $\gamma$. You might find people who argue with the first point, but I don't think you'll find many who argue with the second

I'll move on to trying to give a less formal presentation of the meaning of these rather abstract statements. Let's start out with an example of a non-curved 2d space, a flat 2d plane, and a curved 2d space, the surface of a sphere. Due to the complexity of describing curvature, I won't say how we define curvature in general, but for the example its sufficient to have one curved space and one non-curved space.

Note again that these surfaces are entirely spatial surfaces. It is not space-time curvature we are talking about in this particular context, just spatial curvature. Now we can study how light propagates in these examples. If we emit light signals from a point on the planar surface, it spreads out in all directions, the light rays never meet again. If we emit light from a point on the surface of a sphere, we need to know how it propagates. We will say that it follows the shortest path between two points, (which is a great circle on a sphere). Then we conclude that the light beams on the sphere initially diverge, but, as observed by observers on the sphere, stop diverging and eventually refocus at the antipode.

The effect is a dynamic one. Two stationary observers on the plane, or on the sphere, experience no forces in this example model. They simply lie at rest in the curved space. The apparent "deflection" of light that causes it to refocus in the case of the sphere is just the behavior of light following the shortest distance between two points. It's a consequence of geometry, not "forces". Something similar happens in GR. We observe "extra" deflection due to the purely spatial part of the curvature that we don't see when two observers are at rest relative to each other.

To make the argument airtight it really requires mathematics. Hopefully this overview provides more insight than confusion.

I will say a bit more about the topic, though. One of the differences between spherical geometry and planar geometry is the excess of angles of triangles. The sum of the angles of a triangle is always 180 degrees on a plane, but on a sphere, it can be more. Further, the amount more depends on the area enclosed by the triangle. The theoretical predictions of GR share similar features, though our experimental testing doesn't attempt to measure the sum of the angles of a triangle. Conceptually, though, the prediction of GR is that spatial geometry is not Euclidean. If we take 2d slices of space (not space-time), we'd expect that depending on the Gaussian curvature of said spatial slice, that the sum of angles of a large triangle would be different than 180 degrees. And it would also hold true that the magnitude of the effect would depend on the size of the triangle. I don't think I've seen anyone actually calculate these effects, though.

Gaussian curvature <<link>> is a good way to move up to start to study of how curvature is defined. For a 2d surface, the Gaussian curvature is an intrinsic measure of curvature, and is represented by a single number. The formulation of the more general curvature tensors can be thought of as giving the Gaussian curvature of any selected plane, and knowing the Gaussian curvavature of all possible plane slices gives us sufficient information to reconstruct the curvature tensor. It's more productive to focus on the Gaussian curvature as an intrinsic property of the 2d space, though the Wiki article starts out by describing and defining it in term of so-called "extrinsic" curvature :(.

 

[PeterDonis]

it seems reasonable to conclude that the "extra" bending of light can be ascribed to the curvature of space, more formally the PPN parameter $\gamma$.

If you adopt the PPN framework, yes. But as you note, that framework is limited to weak gravity. It won't explain, for example, the orbits of light rays close to a black hole.

Also, the PPN framework requires a particular choice of coordinates (for the solar system, these are basically coordinates in which the barycenter of the solar system is at rest). Only in those coordinates is "space curvature" a reasonable term for what is producing the "extra" bending of light over the "Newtonian" prediction. (Which, as I've already noted, is not really made using Newtonian gravity, but using the equivalence principle in a local inertial frame, which means using SR in a local inertial frame.)

 

[Dale]

the bending is mainly depending on the speed of the very same particle (neutrino) and not on the (supposed) curvature.

You make an interesting point here. However it is important to understand that the curvature is curvature of spacetime, not just space. Different velocities are different directions through spacetime. So you can think of it somewhat like hitting a speed bump head-on or at an angle. The curvature of the speed bump is the same, but the angle of the path makes a difference.

 

[vanhees71]

I'm not sure about the historical facts although nowadays it's easy to study the development of GR due to the great work of the Einstein Paper Project. There have been also very detailed science-historical studies on this topic.

As far as I understand it, the first prediction by Einstein (and Grossmann) on the bending of light due to gravity was wrong by a factor of 2, because they used a naive non-relativistic limit of the then unknown Schwarzschild metric. In the standard coordinates it reads
$$\mathrm{d} s^2=\left (1-\frac{r_s}{r} \right) \mathrm{d} t^2 - \left (1-\frac{r_s}{r} \right)^{-1} \mathrm{d} r^2 - r^2 (\mathrm{d} \vartheta^2+\sin^2 \vartheta \mathrm{d} \varphi^2).$$
The leading-order non-relativistic limit delivers only the correction of the temporal part of the pseudo-metric, and as far as I remember Einstein and Grossmann just took the spatial part without the term in front of $\mathrm{d} r^2$.

Then, in 1914 the excursion of Erwin Freundlich to watch the solar eclipse couldnt' be undertaken, because of the outbreak of World War I. That was good for Einstein since his prediction of the light-bending effect was too small by a factor 2, and it was the more impressive that Edington in 1919 found agreement with the then known correct prediction of the full theory of 1915, so that Einstein became the first "shooting star" of theoretical physics ever.