# Light Cone in Rindler Coordinates: Visualization & Reasoning

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• kent davidge
In summary, the conversation discusses the visualization of light cones in Rindler coordinates and the possibility of null geodesics crossing the Rindler horizon. The speaker provides a drawing and reasoning, while also mentioning a derivation of the null geodesics on Wikipedia. They also mention that outgoing null geodesics can't cross the Rindler horizon, but ingoing ones can, although this won't be visible in Rindler coordinates. The speaker also clarifies that the equations used are not wrong and are equivalent to solving the geodesic equations for null geodesics. They also discuss the concept of events that will never be in the past light cone of the accelerating observer and compare it to the event horizon of a black hole
kent davidge
Im trying to visualize what form the light cones take in Rindler coordinates. Below is my drawing + reasoning. Is it right?

It looks ok to me. Wiki has a derivation of the null geodesics in <<link>>, but it's unnecessarily complex for what you want to do.

p=0 is a coordinate singularity, usually called the Rindler horzion, because the metric coefficient of dn^2 vanishes. I don't believe that the null geodesics should be able to cross the Rindler horizon, and in your diagram they don't.

pervect said:
Wiki has a derivation of the null geodesics in <<link>>
So it seems like I have obtained the correct diagram from wrong equations.
It seems that I should have considered the geodesic equations first.

pervect said:
I don't believe that the null geodesics should be able to cross the Rindler horizon

"Outgoing" ones (the ones going in the same direction as the Rindler observers' proper acceleration) can't, but "ingoing" ones (the ones going in the opposite direction) certainly can. However, this won't be visible in Rindler coordinates because they don't cover the Rindler horizon; instead, an ingoing null geodesic will look like it asymptotically approaches ##p = 0## as ##n \rightarrow \infty## but never reaches it.

Also, the above is assuming that we are looking at where geodesics go, but we also need to look at where they come from. That's just the time reverse of the above: outgoing null geodesics can come from below the Rindler horizon while ingoing null geodesics can't. An outgoing null geodesic will asymptotically approach ##p = 0## in Rindler coordinates but never reach it as ##n \rightarrow - \infty##.

vanhees71
kent davidge said:
So it seems like I have obtained the correct diagram from wrong equations.
It seems that I should have considered the geodesic equations first.

Your equations aren't wrong. What you did, for this simple case, is equivalent to solving the geodesic equations for null geodesics. In the Wikipedia article, that corresponds to setting ##P = Q = 0## and ##y = z = 0##.

In the more general case where we put back the other two spatial dimensions, what you did is equivalent to solving the restricted set of null geodesic equations that only apply to "radial" geodesics--geodesics that only move in the ##n - p## plane, not in the other two coordinate directions.

vanhees71
PeterDonis said:
Your equations aren't wrong. What you did, for this simple case, is equivalent to solving the geodesic equations for null geodesics. In the Wikipedia article, that corresponds to setting ##P = Q = 0## and ##y = z = 0##.

In the more general case where we put back the other two spatial dimensions, what you did is equivalent to solving the restricted set of null geodesic equations that only apply to "radial" geodesics--geodesics that only move in the ##n - p## plane, not in the other two coordinate directions.
Ah, ok. What seems weird to me is that if you let ##n \rightarrow \infty## the past light cone of the particle will cover only half of the space. Is that right? So it seems that there are events that will never cross the particles world line.

kent davidge said:
So it seems that there are events that will never cross the particles world line.
Yes, there are events that will never be in the past light cone of the accelerating observer. That’s the Rindler horizon at work.
[Edit: “past light cone of the accelerated observer” is sloppy wording. “Past light cone of any event on the worldline of the accelerated observer” might be better]

Compare with the event horizon around a Schwarzschild black hole: events on the inside of the horizon will never be in the past light cone of an observer outside.

Last edited:
kent davidge and vanhees71

## 1. What is a light cone in Rindler coordinates?

A light cone in Rindler coordinates is a visualization of the paths that light rays take in a spacetime diagram using a specific coordinate system known as Rindler coordinates. It is used to understand the relationship between space and time in a flat, accelerating universe.

## 2. How is a light cone in Rindler coordinates different from a light cone in Minkowski coordinates?

A light cone in Rindler coordinates is different from a light cone in Minkowski coordinates because it takes into account the effects of acceleration and non-inertial frames of reference. This allows for a more accurate representation of the paths of light rays in a non-uniform universe.

## 3. What does the visualization of a light cone in Rindler coordinates tell us?

The visualization of a light cone in Rindler coordinates tells us about the causal structure of spacetime, specifically the relationship between events that can be influenced by each other. It also helps us understand the effects of acceleration on the paths of light rays.

## 4. How is a light cone in Rindler coordinates used in scientific research?

A light cone in Rindler coordinates is used in various fields of physics, including general relativity and cosmology. It is used to study the effects of acceleration on the behavior of light and other particles, as well as to understand the structure of spacetime in non-uniform environments.

## 5. Can a light cone in Rindler coordinates be used to visualize other phenomena besides light?

Yes, a light cone in Rindler coordinates can be used to visualize the paths of any type of particle or object in a non-uniform universe. This includes not only light, but also matter particles and even entire galaxies. It is a useful tool for understanding the behavior of objects in accelerating and non-inertial frames of reference.

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