What is the shape of the Rindler horizon?

Question

Consider a relativistically accelerating body, it is said that from its frame of reference there would appear to be a Rindler horizon at a distance $d =\frac{c^2}{a}$. My question is about the body's perceived shape of the Rindler horizon. Would the Rindler horizon simply look like a black, infinite 2D plane, perpendicular to the body's direction of acceleration? If so does it appear instantaneously at the moment $t=0$ when the body starts to accelerate, or does it smoothly evolve over time, say starting from the shape of a sphere, approaching the shape of an infinite 2D plane at $t \rightarrow \infty$. While it does make sense that along the body's direction of acceleration, the Rindler horizon would be at a distance $d =\frac{c^2}{a}$, I don't think it makes sense that the rest of the Rindler horizon would lie on the same plane (And therefore be a 2D plane), because wouldn't the light coming from the directions which are not perpendicular to the body's direction of acceleration, have to travel a larger distance to reach the body? $$$$I tried searching online for some information regarding this topic, but I couldn't find any clear answer regarding the shape of the Rindler horizon. Any insight would be greatly appreciated!

Answer 1

You would not "see" a horizon. While it is true that, say, when you hit the gas in your car and accelerate forward at a rate of "0 to 60" in 5 seconds, a Rindler horizon appears about 17 Pm behind your ass as measured in the Rindler frame, which is closer than the nearest foreign star, Proxima, at 41 Pm distant, if you are looking out the back window at the time the car accelerates, you won't suddenly see the stars disappear even though all of them will be behind it. That's because photons near you are still carrying information regarding what was going on at those times there long in the past before you hit the gas pedal (and before your standard frame of reference could be understood as a Rindler frame).

(If you want to understand the significance or meaning of the Rindler coordinates, or reference frames in general, that's a different question.)

The Rindler horizon is a surface in space-time. It is not proper to speak of its "shape" in a visualizable form, without choosing a particular method of perception or visualization, that separates space and time. I presume you mean in the Rindler frame just mentioned, which is where that distance figure comes from. But you should note this, like the Lorentz frames on which it is based, is not the frame representing the field of view of your eyes and, indeed, depending on how you interpret it, can be considered as either not to represent a real physical observer at all, or else, one that "senses" in a way vastly different from how your eyes do but is useful for mathematical reasons. In that Rindler frame, yes, the horizon is indeed an infinite flat plane at the distance just mentioned. But calling it "black" is to assume that you can see it. You can't, and won't.

Instead, what you will see if you could accelerate for extremely long times at a constant rate - i.e. replace your car, airplane or other humble vehicle with a giant bolus full of anti-matter and a throttle to control it - as your relative speed approaches $c$, the things in the direction retrograde to your motion will look increasingly redshifted and dimmer, and those in the prograde direction will look increasingly blueshifted and brighter. Moreover, that blueshifted view will tend to concentrate on the prograde point on your view sphere, becoming very bright and very hot - eventually, bright & hot enough to wound and kill you. This phenomenon is called "relativistic beaming", and importantly, it will stay even once you take your foot off the "gas" and continue to coast through space. That is because this is purely an effect of velocity, not acceleration.

While accelerating, the Rindler horizon will "appear" to you in the form of the increasing time dilation and redshifting of the stars behind you "conspiring" to prevent you from receiving information from Rindler-inaccessible events on their world lines so long as that acceleration continues.

Answer 2

The shape of the Rindler horizon can easily be understood in the inertial frame. The past and future Rindler horizons are two null half planes. The 3D null sub manifolds formed by those planes have no Riemann curvature. So they are flat.

You can also take a slice of the horizon at a fixed moment in time. That gives you a 2D spacelike sub manifold. That sub manifold also has no Riemann curvature, regardless of whether your simultaneity is defined in the Minkowski frame or the Rindler frame. So again, they are flat.

wouldn't the light coming from the directions which are not perpendicular to the body's direction of acceleration, have to travel a larger distance to reach the body?

Yes, but the same can be said of any flat Euclidean plane in an inertial frame. So this is not a valid objection.