# How do Rindler coordinates fit into special relativity?

It is often said that special relativity doesn't handle acceleration since you general relativity for that. If I understand correctly this is incorrect just with some caveats.

Rindler coordinates are coordinates as experienced by a constantly accelerating observer such that the observer is at rest in that frame. They are given by

$$\begin{array}{ccc}T=x\sinh\alpha t&\leftrightarrow&t=\frac 1\alpha\text{arctanh}\frac T X\\ X=x\cosh\alpha t&&x=\sqrt{X^2-T^2}\end{array}$$ Where $$X,T$$ are the coordinates in the lab frame and $$t,x$$ are the coordinates of the accelerated observer. $$\alpha$$ is the proper acceleration.

I have the following questions about these coordinates

1. Is the space according to the accelerated observer curved? If so why not? Isn't it similar to a uniform gravitational field in the $$x$$ direction?
2. Is this a valid frame? I mean valid in the sense that in Newtonian mechanics acceleration frames are not valid because they experience ghost forces.
3. When do you need GR to handle acceleration and when can you get away with using SR?

This can only be answered by pointing out the difference between special and general relativity. There is the historically motivated definition, which is still in widespread use in popular and semi-popular accounts, according to which special relativity only deals with inertial frames and coordinates (1a), while general relativity deals with accelerated frames and coordinates (1b). According to this definition, Rindler coordinates belong to general relativity.

However, physicists realized that a distinction between theories by a certain choice of coordinates doesn't make much sense, because the content of a physical situation cannot be dependent on the choice of coordinates by which we describe that physical situation. A perfect example is the case of constant proper acceleration – it produces hyperbolic motion when we describe it in terms of inertial frames (so we are in the realm of special relativity by definition [1a]), whereas the comoving frame in hyperbolic motion is nothing other than the Rindler coordinate frame (following definition [1b] we are suddenly in the realm of general relativity, even though we are still describing the same physical situation).

All of this shows that we need a definition that is actually based on a difference in physics, not only on a difference in coordinates. Fortunately, such a distinction is possible using the concept of spacetime curvature: We can distinguish between "flat" spacetimes in which spacetime curvature is zero (such as Minkowski metric, Rindler metric, Born metric, etc), versus "curved" spacetimes (such as Schwarzschild metric, Friedmann–Lemaître–Robertson–Walker metric, etc.) which has its source in the stress-energy tensor.

Note that "curved spacetime" is not the same as "making space appear curved by using curvilinear coordinates": Spacetime curvature manifests itself in real, non-uniform gravitational fields and tidal gravity and is independent of the choice of coordinates, whereas the curvilinear space coordinates in terms of Rindler coordinates and its related "uniform" gravitational field is only an artifact of the coordinates used and can always be transformed away by switching back to inertial coordinates. This finally leads to the definition:

(2a) Special relativity is the theory of flat spacetime (it includes inertial frames, Rindler coordinates, Born coordinates, "uniform" gravitational fields, etc.)

(2b) General relativity is the theory of curved spacetime with the stress-energy tensor as the source of all gravitational effects (it includes non-uniform gravitational fields and tidal effects, thus all “real” gravitational effects independent of coordinates).

• This is really clear. Would this mean that if I calculated the Ricci tensor of the Rindler coordinates it would yield $R_{\mu\nu}=0$? Sep 20 '20 at 14:30
• Yes, see this post and its answer: physics.stackexchange.com/questions/364022/… Sep 20 '20 at 15:25
• I guess I've been living in a bubble: I didn't know anybody ever used inertial/non-inertial frames as the distinction between special/general relativity, neither historically nor in popular articles. For my cultural education, do you happen have any links to sources that use the terms that way? Sep 20 '20 at 16:54
• If anyone would like to read more on this there are a few sources but this point is discussed for example in 'Classical Theory of Fields' by Landau and Lifshitz.
– Tom
Jan 25 at 0:43

I liked Batiatus' answer where we distinguish between flat spacetimes and curved ones, I feel like that gives you a good answer of questions (1) and (2). To answer question (3) I would question this premise that special relativity doesn't “do” acceleration, because I believe that that's the only thing it does.

So a remarkable thing to my mind is that the Lorentz transform can be reconstructed as the limit of itself truncated to first order, $$\lim_{N\to\infty} \begin{bmatrix} 1&-\alpha/N&0&0\\ -\alpha/N&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix}^N=\begin{bmatrix} \cosh\alpha&-\sinh\alpha&0&0\\ -\sinh\alpha&\cosh\alpha&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix}.$$ In a way, this means that what special relativity is “known for”—time dilation and length contraction—are not what it is really “about.” Those are second order effects that come directly from the first order effect, they are consequences rather than new physics in themselves. The new physics is the relativity of simultaneity, $$\gamma$$ is a footnote, a normalization that Nature happens to perform for us to keep the relativity of simultaneity mathematically consistent.

Furthermore these first order boosts happen directly when we accelerate. What it is saying is that when you accelerate, clocks in front of you by a coordinate $$x$$ appear to tick anomalously faster (than say Doppler would suggest), and clocks behind you slower, by a factor of $$1+ax/c^2$$. That is the new physics we are talking about here in special relativity.

The choice to use differential geometry to describe the situation is orthogonal. You can phrase the above as Rindler coordinates if you like, using differential geometry in special relativity. But of course you do not have to. You can also just calculate directly, for example, that there is an event horizon at coordinate $$x=-c^2/a$$. After all I just told you that the clocks aren't ticking there. But if you want to see that event horizon in Rindler coordinates, don't let me stop you!

Similarly, while differential geometry has been the most successful approach to general relativity and I wouldn't want to supplant it, it is not necessary in order to get your bearings there. At least, not in the case of static situations The essential static physics of general relativity is the equivalence principle, which combines with special relativity in a pretty straightforward way. If you wanted to predict that gravitational waves exist but require a varying quadrupole moment of mass you might require more than just the equivalence principle, you might require the full Einstein equations and differential geometry.

But the equivalence principle is enough to conclude that GPS satellites will be seen by us on the ground, who by the equivalence principle are accelerating toward space, as ticking faster than us—real gravitational time dilation. And it is enough to conclude that there may be places in the universe which are black holes, that event horizon that we saw before becoming an essential part of a location in space via the equivalence principle. We just don't see it because our acceleration $$g$$ satisfies $$c/g\approx0.97\text{ years}$$ and so this wall of death should be about a light year underneath our feet, but Earth is much smaller than a light year. You can understand a bit without invoking the full differential geometry approach.

So in some sense, acceleration is the only thing that special relativity does, and you can always get away with using special relativity to describe acceleration. Indeed, to my mind the resistance to talking about acceleration is why people think the twin paradox is a thing. If you know that this is the core of the physics, the twin in space is accelerating towards their sibling on Earth which is a long distance away, so they see the Earth twin age extremely fast. So much for the paradox.

Is this a valid frame? I mean valid in the sense that in Newtonian mechanics acceleration frames are not valid because they experience ghost forces.

You do experience a "ghost" force in Rindler coordinates; there will appear to be a force opposite the direction of acceleration (much like when a car accelerates, objects in the car appear to have an acceleration backwards). But that doesn't exactly mean it isn't "valid". In the most general sense, a coordinate system is simply a method of assigning a 4-tuple of real numbers to each event in space-time. There are properties that are "nice" to have, such as "straight lines" in coordinate space corresponding to geodesics in space-time or there being no singularities, but those aren't required.