Does 'special relativity + newtonian gravity' predict gravitational bending of light?

Question

It seems to me that special relativity (SR) already predicts that gravity will bend light rather than this effect being a proof of general relativity (GR). Photons have energy proportional to their frequency and according to $E = mc^2$ they also have a tiny, but non-zero relativistic mass equivalence. I have read the semantic argument that gravity deals with invariant or rest mass, but this should apply to a hypothetical photon at rest, not to real photons at velocity $c$.

I have considered the possibility that the SR effect is much smaller than the GR effect. GR explains the equivalence of inertial mass and gravitational mass as inherent rather than being a puzzling coincidence, but it is true in Newton's gravity, SR, and GR, so the quantitative difference between GR and SR does not seem right.

Answer 1

Yes, it does, or to be more precise, Newtonian gravitation predicts photons will be deflected if you assume photons have some mass. However the amount of this deflection is just half of what GR predicts. And the observed amount of deflection is what GR predicts (within experimental error).

In particular the way this is studied is by using something called the Parameterized Post-Newtonian (PPN) framework. This is discussed on this Wikipedia page and also in the paper by Will I mention below. PPN is essentially Newtonian gravitation with a bunch of first-order corrections from GR added to it, controlled by various parameters, so it's useful for experimental tests of GR, and comparisons between GR and other metric theories of gravity, where the field is weak: it would not be useful for instance, for tests involving black hole collisions where the field is very much not weak!

I believe the first, simplified, PPN framework was derived by Eddington, specifically for the purpose of understanding how the deflection of light by the Sun differed between GR and Newtonian gravitation.

PPN has a significant number of parameters, but for the case of the deflection of light by a spherically-symmetric field only one matters, which is known as $\gamma$. The angle of deflection is then given (remember this is a first-order approximation which is valid for a weak field) by

$$\delta\theta = \frac{1+\gamma}{2}\frac{4 M_\odot}{d}\frac{1 + \cos\Phi}{2}$$

where $d$ is the distance of closest approach to the Sun, $M_\odot$ is the mass of the Sun, and $\Phi$ is the angle between Earth-Sun line and the incoming-photon line.

In this expression, Newtonian gravitation would say that $\gamma = 0$, and GR would say $\gamma = 1$. So you can see that GR predicts exactly twice the deflection that Newtonian gravitation predicts. And this is what Eddington et al measured on the 29th of May, 1919, and discovered that $\gamma = 1$ (to within a fairly large uncertainty at the time, but it was clear that $\gamma = 0$ was ruled out): this made Einstein famous.

There is a discussion of this in the Wikipedia article on tests of GR, and the paper by Clifford M. Will has more detail I think (disclaimer: I have not checked the latter in detail: it looks like section 3.4 & specifically 3.4.1 may be what you want). My expression for $\delta\theta$ above is lifted from the paper by Will.

Answer 2

As mentioned in tfb's answer, the deflection treating photons with an "effective mass" is half that of general relativity. While photons have zero rest mass in special relativity, one can consider them with infinitesimal mass for the purposes of a Newtonian approximation (a "gravitational force" interacting directly and instantaneously between two masses). Consider a single photon passing a body of mass M at a distance r. The Newtonian deflection angle is: $$\theta= \frac{2GM}{rc^2}$$

In general relativity, the deflection angle is: $$\theta= \frac{4GM}{rc^2}$$ This is because general relativity considers spacetime warpage.

One interesting feature to point out is that in the Newtonian approach, the passing object will experience an increase in speed (potential energy converted into kinetic energy). However, according to relativity, a photon cannot be accelerated past c. Instead, the transfer of this energy causes the photon's frequency to increase as it passes through a gravitational well (and decrease upon escaping); this is the phenomenon of gravitational redshift. The energy of a photon is given by $e=h\nu$, where h is Planck's constant, and $\nu$ is the frequency, thus it can be seen that energy and frequency are directly proportional with Planck's constant as the constant of proportionality.

Answer 3

Let me cast this in a historical perspective.

As Maxwell's equations came into acceptance there was concensus that light is a form of wave propagation. Maxwell's theory of electromagnetism offers an explanation of how light can carry energy.

In addition, it is already implied in terms of Maxwell's equations that electromagnetic radiation carries momentum in the direction of propagation. However, there was no reason to attribute mass to electromagnetic radiation. Hence there was no reason to expext that gravitation would have any effect on light.

Historically, gravitation was thought of as a force that acts instantaneously over any distance. It was necessary to think of gravitation as acting instantaneous, this had been demonstrated by Laplace. If gravitation would propagate at a finite velocity then there would be aberration effects, and none of those were observed.

Changes introduced by Special Relativity.

Let me first discuss gravitation.
The first to explore the consequences of relativistic physics for gravitational theory was Poincaré. (In 1905, the same year that Einstein's paper on Special Realativity came out.) Poincaré noted that if it is assumed that all theories in the area of mechanics must be Lorentz invariant then a new theory of gravitation is necessary, as an infinite speed of gravity is no longer a possiblity. This new theory of gravitation must reproduce the predictions of Newton's law of gravitation for the known celestial mechanics. Poincaré gave some suggestions on how to develop such a Lorentz invariant theory of gravitation.

About electromagnetic radiation:

Indeed as early as 1905 Einstein had offered a consistency argument that in terms of Special Relativity it is necessary to attribute inertial mass to electromagnetic radiation.

That is: Einstein demonstrated that without attributing inertial mass to electromagnetic radiation you get a self-contradiction. So that is logical implication.

The question that you raise is: does that also imply that we should attribute gravitational mass to electromagnetic radiation? You submit: for matter there is no known exception to equivalence of inertial and gravitational mass.

Here, while a suggestion is there, there's no logical necessity to attribute gravitational mass to electromagnetic radiation. So no: Special Relativity does not imply that gravitation must have an effect on light.

But yeah, there is that undeniable suggestion that gravitation should affect light, and as we know the assumption of universal equivalence of inertial and gravitational mass was among the most important guidances as Einstein struggled to develop General Relativity.

GR replaced SR, and the shift from SR to GR was as profound as the shift from Newtonian mechanics to SR. A fundamental assumption of SR is that the Minkowski spacetime itself is an unchanging, static entity. Overthrowing that: in terms of GR spacetime is not static; there is curvature of spacetime, in response to presence of mass/energy.

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Summary:
Logically, Special relativity does not imply gravitational effect on light.

• Special Relativity invalidates the instantaneous-over-distance assumption that is necessary for newtonian gravitational theory.
• Logically, universal equivalence of inertial and gravitational mass is a separate assumption.

Answer 4

(Hi Cyril, welcome to Physics.SE!) You already received a couple good answers here but these mostly deal with Newtonian physics and don't address your actual question/claim, namely that special relativity should predict the bending of light. (Which is why one could indeed say that the other answers actually answer an "edited question", as Dale puts it in his comment to your question.) So I think it is in order to also point out to you why your arguments involving special relativity (SR) don't work and SR indeed does not predict the bending of light.

according to $E=mc^2$ [photons] also have a tiny, but non-zero mass equivalence

Photons do not have a rest mass and the energy-momentum relation $E=mc^2$ you cited holds only for massive particles at rest. In particular, it doesn't hold for photons. The correct and full energy-momentum relation (for any particle in any frame) reads $E^2=(mc^2)^2 + p^2c^2$, where $p$ is the momentum and $m = 0$ for a photon (and any other massless particle).

this should apply to a hypothetical photon at rest, not to real photons at velocity c

In SR there is no frame in which a photon would be at rest. This is because an object moving at the speed of light in one frame will move at the same velocity in all other frames. In order to understand this a bit better, I recommend you take a look at the book "Relativity, Groups, Particles" by Sexl & Urbantke which presents a particular nice derivation of the laws of special relativity (i.e. the Lorentz transformations) from a small number of reasonable assumptions (transformations between inertial frames be linear et cetera). Starting from these assumptions they arrive at a point in their derivation where they end up with a free parameter (I think they called it $k$) which they can choose at will. It turns out that if this parameter $k$ were chosen to be zero, Galilean mechanics would follow. A non-zero $k$, however, would imply that there is a "universal" velocity $c$, related to $k$ via $c^2 := 1/k^2$ (as far as a I remember), which is universal in the sense that any object traveling with velocity $c$ in one (inertial) reference frame would appear to be moving with $c$ in all other (inertial) reference frames, as well. Needless to say, experiments dictate that $k$ should indeed be non-zero, as we observe light to travel at the same velocity in every reference frame. The result of Sexl & Urbantke's derivation for non-zero $k$ are then the famous Lorentz transformations of special relativity. It turns out that these transformations leave the so-called Minkowski metric and, in particular, every particle's 4-velocity and 4-momentum invariant. It is from these structures that all other relations (like the energy-momentum relation and the invariance of mass$^†$) follow. In particular, it follows that photons cannot have a mass.

To summarize: Once you accept the fact that 1) there is a universal velocity $c$ which is the same in all reference frames and that 2) photons move at this universal velocity $c$, you are immediately forced to conclude that photons cannot have a non-zero mass and that there cannot be a reference frame in which they are at rest. So your remaining argument that they should be subject to gravitation in a theory of special relativity (with Newtonian gravity) does not work.

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†) Here I assume that "mass" is defined as the Minkowski norm of the particle's 4-momentum $p$, $m^2 := -p_\mu p^\mu$. The fact that this definition makes sense again follows from comparing predictions with experimental observations.

Answer 5

It's really trivially easy (even if ultimately trivially wrong) to conclude that gravity has an effect on massless objects, using purely Newtonian mechanics - even without special relativity.

If:

$$F = \frac{GMm}{r^2}$$

and

$$F = ma$$

then

$$ma=\frac{GMm}{r^2}$$

$$a=\frac{GM}{r^2}$$

where M is simply the mass of the larger body (e.g. the star) and r is the distance of the smaller body from the center of mass of the larger one. The mass m of the smaller object has no meaningful effect on its own acceleration toward the larger one, including when m = 0. In other words, anything whose mass is negligible compared to a massive object, given any particular distance r from the massive object, should be accelerated toward the massive object's center of mass regardless of its own mass or lack thereof.

This is the same simple math that gives us the fact that acceleration due to gravity at the Earth's surface is 9.8m/s^2 when the mass of the falling body is assumed to be negligible relative to the mass of the Earth.

Again, we know this formula doesn't hold under General Relativity (for light it's actually off by a factor of 2), and thus it's wrong, but anyone studying this problem pre-Einstein - including, one assumes, Einstein himself - would likely have concluded that light is affected by gravity whether it has mass or not.

It seems people in Newton's time assumed this meant that light must have mass, though I would personally wonder why, given that the existence of convergent limits are a fundamental assumption of calculus and a zero-mass particle being deflected by mass is thus perfectly harmonious with the math Newton himself co-invented. I guess back then they were less willing to just accept the math when it conflicted with intuition than we are today.