Derivation of Rindler Metric and How It Resolves the Twin Paradox

[Chare]

From what I have read the twin paradox can be resolved with the Rindler metric and without the need to bring in general relativity. Special relativity will suffice. But how does the Rindler metric get derived in the context of a constant accelerating reference frame. I haven't seen anything in my searches that answers the below questions in a clear manner.

To be more precise about what I mean, one way the time dilation is seen in non-accelerating reference frames is by considering for example light particles bouncing between two nearby mirrors in a clock and thus concluding moving clocks time slower. These thought experiments give us the Lorentz transformations a way of translating events (t, x, y, z) between two reference frames in a bijective mapping.

But what is the derivation of the Rindler metric. It presumably gives a way to map events (t, x, y, z) to (t', x', y', z') in another reference frame where one reference frame is accelerating at a constant rate. But how does one justify whatever details about it. There should be some kind of thought experiment with light bouncing between mirrors.

But moreover its not even clear to me how does the accelerating reference frame speak of time. Won't they have problems synchronizing clocks in their reference frame (which is accelerating).

 

[Ibix]

Dolby and Gull's paper on radar time (https://arxiv.org/abs/gr-qc/0104077) provides a fairly general method for producing coordinate systems. If you use a radar set undergoing constant proper acceleration, I believe you'll get Rindler coordinates (edit: having now re-read the paper - yes you do). Constant speed gets you the usual Einstein coordinates. They explicitly provide simultaneity planes for the traveling twin in the twin paradox, for both an instantaneous and "slow" turnaround scenario.

By the way, there is never any need to introduce General Relativity to resolve the twin paradox. The resolution is simply that the proper time between two events depends on the path taken between them, just as the distance between two points in space depends on the path taken from one to the other. You do need curved coordinate systems (such as Rindler coordinates) if you wish to describe the perspective of the moving twin, but (a) this isn't necessary to resolve the paradox; and (b) the situation is still in flat spacetime so in the domain of Special Relativity.

 

[Orodruin]

It is worth pointing out that, while the world lines of constant spatial coordinates in Rindler coordinates correspond to world lines with constant proper acceleration, the proper acceleration of each of those world lines is different. That being said, as has already been pointed out, there is nothing prohibiting you from using standard Minkowski coordinates to resolve the twin paradox. In fact, it is only a "paradox" if you do not understand special relativity properly - in particular the relativity of simultaneity. If you can accept that the straight line is the shortest line between two points in a plane, then you essentially have all of the ingredients you need to understand the geometry that underlies the differential ageing in the twin "paradox". Also see my PF Insight on this.

 

[PAllen]

Actually @Ibix , radar coordinates for a uniformly accelerating observer match simultaneity of Rindler coordinates, but they don’t match distances. See, for example, :

https://en.m.wikipedia.org/wiki/Rindler_coordinates#Notions_of_distance

 

[stevendaryl]

From what I have read the twin paradox can be resolved with the Rindler metric and without the need to bring in general relativity. Special relativity will suffice. But how does the Rindler metric get derived in the context of a constant accelerating reference frame. I haven't seen anything in my searches that answers the below questions in a clear manner.

I wrote two Insights articles about the subject. To make a long story short, the Rindler coordinate system is the most natural coordinate system for passengers on board an accelerating spaceship.

1. Assume the rear of the spaceship is traveling in such a way that the proper acceleration is constant. Proper acceleration is the acceleration as measured in any inertial frame in which the spaceship is momentarily at rest.
2. Assume that the various parts of the spaceship are undergoing what's called Born rigid motion. What that means is that the proper distances between any two parts is constant. Once again, proper distance is the distance as measured in the inertial reference frame in which the spaceship is momentarily at rest.

Those two assumptions define Rindler motion for the spaceship. For a spaceship undergoing Rindler motion, you can set up a coordinate system appropriate for passengers on board the spaceship as follows:

1. Define the coordinate $R$ for parts of the spaceship to be just the $x$-coordinate that the part had at the time of launch.
2. Declare two events that take place on board the spaceship to be "simultaneous" if they are simultaneous according to the inertial reference frame. That is, two events $e_1$ and $e_2$ are simultaneous if there is an inertial reference frame in which they are simultaneous, and in that reference frame, the spaceship is momentarily at rest at the time when those events take place.
3. Having defined simultaneity, we pick as a time coordinate $T$ just the time as measured on the rear rocket (or any reference rocket aboard the spaceship). Times at other places on the ship can then be computed by using simultaneity definition #2 to find the time as shown on the reference clock.

These assumptions and definitions uniquely associate a pair of coordinates $(R,T)$ for every event on board the spaceship. You can extend this coordinate system to describe events outside the spaceship, as well (imagine having a bigger spaceship, so that the events take place inside the spaceship), but unlike regular inertial coordinates, theses coordinates cannot be extended to all space and time.

You can work out how $R$ and $T$ relate to the inertial coordinates $(x,t)$ for the "launch" frame just using algebra, calculus and Special Relativity. It's a little tedious, but the above description sketches the intuitive meaning of these coordinates for an accelerating rocket.

 

[Chare]

I wrote two Insights articles about the subject. To make a long story short, the Rindler coordinate system is the most natural coordinate system for passengers on board an accelerating spaceship.

1. Assume the rear of the spaceship is traveling in such a way that the proper acceleration is constant. Proper acceleration is the acceleration as measured in any inertial frame in which the spaceship is momentarily at rest.
2. Assume that the various parts of the spaceship are undergoing what's called Born rigid motion. What that means is that the proper distances between any two parts is constant. Once again, proper distance is the distance as measured in the inertial reference frame in which the spaceship is momentarily at rest.

Those two assumptions define Rindler motion for the spaceship. For a spaceship undergoing Rindler motion, you can set up a coordinate system appropriate for passengers on board the spaceship as follows:

1. Define the coordinate $R$ for parts of the spaceship to be just the $x$-coordinate that the part had at the time of launch.
2. Declare two events that take place on board the spaceship to be "simultaneous" if they are simultaneous according to the inertial reference frame. That is, two events $e_1$ and $e_2$ are simultaneous if there is an inertial reference frame in which they are simultaneous, and in that reference frame, the spaceship is momentarily at rest at the time when those events take place.
3. Having defined simultaneity, we pick as a time coordinate $T$ just the time as measured on the rear rocket (or any reference rocket aboard the spaceship). Times at other places on the ship can then be computed by using simultaneity definition #2 to find the time as shown on the reference clock.

These assumptions and definitions uniquely associate a pair of coordinates $(R,T)$ for every event on board the spaceship. You can extend this coordinate system to describe events outside the spaceship, as well (imagine having a bigger spaceship, so that the events take place inside the spaceship), but unlike regular inertial coordinates, theses coordinates cannot be extended to all space and time.

You can work out how $R$ and $T$ relate to the inertial coordinates $(x,t)$ for the "launch" frame just using algebra, calculus and Special Relativity. It's a little tedious, but the above description sketches the intuitive meaning of these coordinates for an accelerating rocket.

But how does one explain time speeding up on the Earth from the perspective of the guy on the spaceship as he is accelerating back to return to earth. In the part where the spaceship is going at constant velocity outwards the guy in the spaceship sees the guy on Earth aging slower than him. In the part where the spaceship is going heading back to Earth at constant velocity after the acceleration the guy in the spaceship see the guy on Earth aging slower than him. Since the guy in the spaceship arrives at Earth younger than his twin then he must perceive the guy on Earth to age rapidly while in the acceleration phase. How does one explain this? I understand the slower aging for constant velocity because in a hypothetical light clock the photon has to travel between two mirrors further because the thing is moving and that explains slower time. But when the guy on the spaceship is accelerating back how can he perceive a photon bouncing between two mirrors on Earth speeding up to explain seeing the guy on Earth aging faster, this being compared to his own photon bouncing between two mirrors on the spaceship. The photon's velocity is fixed and can't go faster.

 

[stevendaryl]

But how does one explain time speeding up on the Earth from the perspective of the guy on the spaceship as he is accelerating back to return to earth. In the part where the spaceship is going at constant velocity outwards the guy in the spaceship sees the guy on Earth aging slower than him. In the part where the spaceship is going heading back to Earth at constant velocity after the acceleration the guy in the spaceship see the guy on Earth aging slower than him. Since the guy in the spaceship arrives at Earth younger than his twin then he must perceive the guy on Earth to age rapidly while in the acceleration phase. How does one explain this? I understand the slower aging for constant velocity because in a hypothetical light clock the photon has to travel between two mirrors further because the thing is moving and that explains slower time. But when the guy on the spaceship is accelerating back how can he perceive a photon bouncing between two mirrors on Earth speeding up to explain seeing the guy on Earth aging faster, this being compared to his own photon bouncing between two mirrors on the spaceship. The photon's velocity is fixed and can't go faster.

This is perhaps unsatisfying, but if you have a noninertial coordinate system, there just isn't any physical explanation for why clocks speed up or slow down. If you start with one coordinate system and stretch it or compress it to get another coordinate system, what looks like straight lines in the first coordinate system might look like curved lines in the second. For example, in Cartesian coordinates, if you throw an object in the x-direction at a starting position $y=L, x=0$, then its position as a function of time will be: $x = vt$, $y=L$. That's a straight line, if you plotted x or y as a function of t. But if you switch to polar coordinates $r = \sqrt{x^2 + y^2}$, $\theta = arctan(y/x)$, then the same motion looks like this:

$r=\sqrt{L^2 + v^2 t^2}$
$\theta = arctan(L/vt)$

Plotting $r$ as a function of $t$ or $\theta$ as a function of $t$ does not give straight lines. Viewing the motion using coordinates $r,\theta$, the motion seems accelerated, even though there are no real forces acting on the object.

A similar thing happens when you consider noninertial coordinate systems in spacetime. In an inertial coordinate system, if you have a clock that is traveling inertially, then plotting the time $\tau$ on the clock as a function of coordinate time $t$ will give a straight line. The clock advances steadily. In a noninertial coordinate system, if you plot $\tau$ as a function of $t$, it will not be a straight line. It will seem as if the clock speeds up or slows down. But that's just an artifact of using a noninertial coordinate system.

Clock rates ($\frac{d\tau}{dt}$) are just not particularly meaningful, physically. To get a meaningful notion of "rate", you have to compare the rate of two different processes.

Note: Although the clock rate is not particularly meaningful, it is physically meaningful to integrate it:

$\Delta \tau = \int dt \frac{d\tau}{dt}$

That result will give a meaningful number, which is the elapsed time on the clock. So the time coordinate $t$ and the clock rate $\frac{d\tau}{dt}$ are useful for book-keeping, but that's about it.

 

[Nugatory]

Since the guy in the spaceship arrives at Earth younger than his twin then he must perceive the guy on Earth to age rapidly while in the acceleration phase. How does one explain this?

First, you have to be able to explain it in the simpler case of the classic twin paradox: constant speed outbound and inbound, with zero turnaround time. Start with the twin paradox FAQ, pay particular attention to the section on the Doppler analysis, and be aware that...

I understand the slower aging for constant velocity because in a hypothetical light clock the photon has to travel between two mirrors further because the thing is moving and that explains slower time.

That's not right. One way of seeing this is to consider that as far as someone on the ship is concerned, it's the Earth that is moving at a constant speed so the Earth guy should be the one that is aging more slowly. Time dilation is thus an unhelpful way of taking on the twin paradox. But once you understand where the differential aging comes from in the classic version of the twin paradox, it will be clear you to where I comes from in your more complex acceleration case - the concept is the same, only the calculation is more difficult.

 

[Chare]

This is perhaps unsatisfying, but if you have a noninertial coordinate system, there just isn't any physical explanation for why clocks speed up or slow down. If you start with one coordinate system and stretch it or compress it to get another coordinate system, what looks like straight lines in the first coordinate system might look like curved lines in the second. For example, in Cartesian coordinates, if you throw an object in the x-direction at a starting position $y=L, x=0$, then its position as a function of time will be: $x = vt$, $y=L$. That's a straight line, if you plotted x or y as a function of t. But if you switch to polar coordinates $r = \sqrt{x^2 + y^2}$, $\theta = arctan(y/x)$, then the same motion looks like this:

$r=\sqrt{L^2 + v^2 t^2}$
$\theta = arctan(L/vt)$

Plotting $r$ as a function of $t$ or $\theta$ as a function of $t$ does not give straight lines. Viewing the motion using coordinates $r,\theta$, the motion seems accelerated, even though there are no real forces acting on the object.

A similar thing happens when you consider noninertial coordinate systems in spacetime. In an inertial coordinate system, if you have a clock that is traveling inertially, then plotting the time $\tau$ on the clock as a function of coordinate time $t$ will give a straight line. The clock advances steadily. In a noninertial coordinate system, if you plot $\tau$ as a function of $t$, it will not be a straight line. It will seem as if the clock speeds up or slows down. But that's just an artifact of using a noninertial coordinate system.

Clock rates ($\frac{d\tau}{dt}$) are just not particularly meaningful, physically. To get a meaningful notion of "rate", you have to compare the rate of two different processes.

Note: Although the clock rate is not particularly meaningful, it is physically meaningful to integrate it:

$\Delta \tau = \int dt \frac{d\tau}{dt}$

That result will give a meaningful number, which is the elapsed time on the clock. So the time coordinate $t$ and the clock rate $\frac{d\tau}{dt}$ are useful for book-keeping, but that's about it.

So is this like being on an accelerating bus. You feel a force in your reference frame but at least in typical classical physics the force would be thought of as fictitous, due to being in a non-inertial reference frame. The coriolis force being similarly a fictitious force for another example. And in these examples you could do the calculations in those reference frames but the force would nevertheless still be thought of as fictitious.

So in special relativity the distinction between non-inertial and inertial reference frames is made in regard to speaking meaningfully about things physically.

So then in general relativity where as I understand all reference frames (even accelerating ones) are put on an equal footing and there isn't a distinction about which reference frames are considered physically meaningfully how is this explained?

 

[Chare]

First, you have to be able to explain it in the simpler case of the classic twin paradox: constant speed outbound and inbound, with zero turnaround time. Start with the twin paradox FAQ, pay particular attention to the section on the Doppler analysis, and be aware that...
That's not right. One way of seeing this is to consider that as far as someone on the ship is concerned, it's the Earth that is moving at a constant speed so the Earth guy should be the one that is aging more slowly. Time dilation is thus an unhelpful way of taking on the twin paradox. But once you understand where the differential aging comes from in the classic version of the twin paradox, it will be clear you to where I comes from in your more complex acceleration case - the concept is the same, only the calculation is more difficult.

Both sides see the others aging slower during the constant speed parts I agree. I did not mean to say otherwise.

I've read part way into the faq. The section on pseudo gravitational fields with the quote "The one thing we need here is that time runs slower as you descend into the potential well of a pseudo force field." is precisely what was missing in my understanding of the reference frame of the spaceship. No video I watched on the twin paradox explained the reference frame of the spaceship adequately which is what bothered me.