# Could time dilation bend the path of a photon? Does time dilation have a refractive index?

The main question I am getting at is, does time dilation have a refractive index? What I mean is, if I were to shoot a laser past a black hole, would the laser's path "bend" strictly from time dilation not considering the gravitational effects?

I know that you cannot have the gravitational time dilation without gravity but would there be a way to separate the time dilation effects from the gravitational effects for the sake of the question?

in my previous post

Would time dilation cause a planet to rotate?

its seems mass would be effected by TD which is why I believe light would be as well.

• There is no time dilation in the frame of a freely falling observer, so how could that make his path "bend"? For photons there is no time dilation, at all. A photon only measures distances. – CuriousOne May 8 '15 at 22:02
• ok well then what about the frame of the observer? – Joe May 8 '15 at 22:07
• The frame of which observer? In general relativity there is a six (or is it ten?) dimensional infinity of them. – CuriousOne May 8 '15 at 22:12

The main question I am getting at is, does time dilation have a refractive index? What I mean is, if I were to shoot a laser past a black hole, would the laser's path "bend" strictly from time dilation not considering the gravitational effects?

We don't talk of gravitational lensing for nothing, but IMHO you're getting this back to front Joe. See what John Rennie said? Einstein said something similar, see for example this: the curvature of light rays occurs only in spaces where the speed of light is spatially variable". Now take a look at the Wikipedia Riemann curvature tensor article, and see this depiction of curved spacetime:

CCASA image by Johnstone, see Wikipedia

Imagine that we had a whole load of light clocks, and we dotted them throughout an equatorial slice of space around the Earth. Because the speed of light is spatially variable, the light-clock rates vary. When we plot them, such that lower slower clock rates are depicted as lower down in a 3D depiction, and higher faster clock rates are depicted as higher up, we get the picture above. Time dilation doesn't have a refractive index. Time dilation is there because space has something akin to a refractive index. Because a concentration of energy in the guise of the Earth "conditions" the surrounding space, altering its metrical properties, this effect diminishing with distance. Einstein's exact words were:

"According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials g$_{\mu\nu}$), has, I think, finally disposed of the view that space is physically empty."

Note how Einstein referred to inhomogeneous space? IMHO Inhomogeneous vacuum: an alternative interpretation of curved spacetime is worth a read. People tend to think of curved spacetime as curved space and curved time, but IMHO it's better to think of it as inhomogeneous space where the inhomogeneity or your plot of measurements or your metric is curved. Metric is to do with measurement.

I know that you cannot have the gravitational time dilation without gravity but would there be a way to separate the time dilation effects from the gravitational effects for the sake of the question?

Yes, look at the picture above again. Gravitational time dilation relates to the depth of potential, the force of gravity depends on the slope, and the tidal force relates to how curved it is.

Newcomers to relativity tend to regard concepts like time dilation and Lorentz contraction as somehow fundamental concepts from which relativity it derived, but this is not the case. Time dilation arises because the integral of proper time along the worldline of some object will not necessarily match the coordinate time measured by a distant observer. So to ask if time dilation explains why light bends is a somewhat meaningless question. Both time dilation and light bending are effects that arise from the same cause i.e. spacetime curvature.

However there is a sense in which spacetime, for example around a black hole, does have a refractive index.

Back in the days when you learned geometrical optics you no doubt learned that the reason for refraction is that the speed of light changes in different media (like a glass lens) and this causes a relative phase shift. The phase shift then causes refraction. Well from the perspective of a distant observer the speed of light changes near a massive object. See for example the discussion in Speed of light in a gravitational field?, or a search will find many related questions on this site.

Since the speed of light is changing near a massive object we can interpret this as a change in the refractive index, and we'd expect a change in refractive index to bend the light. So it shouldn't be any surprise that light bends near a massive object.

• I like this answer, but I'd rather say that the "same cause" is simply that different paths have different lengths (proper times), which is what's suggested by the prior sentences in this answer anyway. ... It's conceptually the same even with vanishing curvature. – Stan Liou May 9 '15 at 11:41