Twin Paradox in a Closed Universe
September 21, 2007 – 3:14 amIf you’re familiar with the twin paradox, you can skip straight to the Closed Universe section.
Suppose you’re sitting in a car heading east at 100 km/h, while another car passes you by heading west at 100 km/h. According to special relativity, if you compare your clock with the clock of the passenger in the other car (let’s call her Alice), you will notice that her clock runs slower than yours.
The closer your relative speed is to the speed of light, the more noticeable the difference will be. For example, if your relative speed is 90% the speed of light, you will see Alice’s clock running about 2.3 times slower than yours. Of course, it’s not just the clock that will run slower — everything will run slower, including, for example, biological processes. Alice will age 2.3 times slower than you.
But what makes you so special? Nothing, really. Because according to Alice, it’s you whose aging slowly. So whose right? Well, the short answer is that you’re both right, because time isn’t an absolute thing — it depends on who’s measuring it.
This line of reasoning leads to the famous ‘twin paradox’: Two twins, Alice and Bob, are born on earth (if you must know then yes, it’s the same Alice). At birth, Alice is put aboard a spaceship that can reach extremely high speeds and sent off into space for a long trip. She reaches the edge of the galaxy, makes a U-turn, and returns to earth. This round-trip takes her 40 years in earth-time.
Meanwhile, Bob remains on earth. When Alice gets back, Bob is 40 years old. Because Alice traveled at high speed, Bob saw her aging very slowly, so when she gets back she is much younger than Bob, say 20 years old. But according to Alice, it’s Bob who aged slowly, so actually Alice should be older than Bob.
That’s the twin paradox, but it isn’t really a paradox. The problem is that in order to make this round-trip, Alice has to accelerate — she can’t go away and back again with constant speed. Therefore, the problem is no longer symmetrical: Alice is accelerating, Bob isn’t. To calculate exactly what happens in this case you need to use general relativity, but the answer is that Bob is right and Alice is wrong: Alice will have aged more slowly than Bob.
Closed Universe
I’ve been wondering, what happens to the twin paradox if the universe is closed? What I mean by ‘closed’ is that if you travel in a straight line, you end up back where you started. So the universe looks like a closed loop, only in three dimensions. (*)
In such a universe, the Alice twin can leave earth, travel with constant speed, and return to earth by simply going straight (**). Meanwhile, the Bob twin remains on earth like before. No one is accelerating, so the usual explanation of ‘Alice is accelerating and that’s cheating’ doesn’t apply here. Apparently, the situation is symmetrical: Bob thinks Alice is moving and aging slowly, Alice thinks Bob is. Who is right?
Well the answer is rather interesting. In special relativity we are used to saying that there are no ‘preferred frames of reference’ — that all inertial observers are the same, and that there are no absolute speeds, only relative ones. It turns out that in a closed universe this is no longer the case.
Consider the following experiment. Bob, who is on non-moving earth, simultaneously fires two photons at opposite directions. The photons travel around the universe and return to Bob, both at exactly the same instant.
Now let Alice perform this experiment on her spaceship. She sends out two photons, traveling at the speed of light. Let’s look at this experiment from Bob’s frame of reference, where the photons are still traveling at the speed of light (***). Bob sees the photons going around the universe, but meanwhile Alice is moving. So one photon will reach Alice before the other. Alice must observe the same thing (although they will not agree on the timing).
The same experiment, conducted by Bob or by Alice, produces different results. This is because Bob’s frame is special: It has zero speed, an absolute speed. Thus the symmetry between Bob and Alice is broken. Further calculations are required, but it turns out that Bob is correct: Alice ages less than Bob.
If you want more details, I found an interesting paper that discusses this paradox, and also develops the Lorentz transformations for such a universe.
(*) The mathematical term for this is actually ‘compact’, but that’s because mathematicians enjoy complicating things.
(**) You may be wondering how it’s possible to go in a circle without accelerating. You can try looking at it this way: In circular motion with constant speed, the vector of acceleration points inward toward the center of the circle. If you live in a 2-dimensional world, you can measure this vector because it points in a direction you can move in. For instance, you can hang a ball from a piece of string and watch the string stretch.
But if you live in a closed, 1-dimensional world, and you’re going in a circle, where is the acceleration vector pointing? It can’t point inward, because for you there’s no ‘inward’. In your 1D world there’s only front and back. As a consequence, you can’t measure this acceleration experimentally. And what you can’t measure experimentally doesn’t exist.
By the way, in terms of differential geometry, this is what geodesic curvature is all about: It’s the part of the acceleration that’s due to the shape of the manifold.
(***) This is one of the premises of special relativity — that light travels at the same speed for all observers. If you’re unconvinced it’s still true in a closed universe, consider the following. The metric of a cylinder is the same as that of a plane — a cylinder isn’t curved. Similarly, the metric of a closed, 1-dimensional universe is the same as that of an open one:
The constancy of light speed follows from the metric, and from assuming that the correct transformations (boosts) preserve the metric.
![\frac{\partial f}{\partial x}[j] = \frac{f[j+1]-f[j-1]}{2 \, dx}](/latexrender/pictures/5a99aae890fe9ef5482881ed643463ad.png "\frac{\partial f}{\partial x}[j] = \frac{f[j+1]-f[j-1]}{2 \, dx}")
which are applied to your function f as follows:![F[j] = \sum_{n=-l}^{l} f[j-n] * c_n](/latexrender/pictures/9407c374a365816609eb54c168b72267.png "F[j] = \sum_{n=-l}^{l} f[j-n] * c_n")
. How odd! Debugging further, I found that non-zero derivatives only happened when the initial function f had very large (and constant) values, on the order of 
, and dx was very small, about 
.
and summing, some of the least significant bits are lost, because the numbers don’t all have the same exponent. So even though the calculation is accurate, the limitation of the computer’s double precision causes a loss of data. When the convolution is done, you’re left with a small value that’s not zero. But then to get the derivative you divide by dx, a very small number, and this shoots that small error through the roof. The smaller dx, the larger the derivative of the constant function! So once again, decreasing dx actually caused a larger error.
. Specifically, anti-symmetric coefficients have the same exponent. Therefore I changed the summing order to sum pairs of anti-symmetric factors:![F[j] = \sum_{n=1}^{l} (f[j-n] \cdot c_n + f[j+n] \cdot c_{-n}) + f[j] \cdot c_0](/latexrender/pictures/030368e3f1eca0972f29ac06368059f6.png "F[j] = \sum_{n=1}^{l} (f[j-n] \cdot c_n + f[j+n] \cdot c_{-n}) + f[j] \cdot c_0")
prisoners are planning an escape. They bribed a guard to let them all meet once. Each prisoner was given a task to complete, such as digging or stealing some key. Once all the tasks are complete, the prisoners will notify the corrupt guard who will turn a blind eye while they escape. The problem is, the prisoners can’t talk to each other because they are all in solitary confinement. 
) this: 36 kB/s for download is very little compared with my ADSL line’s capacity of 180 kB/s. But to get the full speed I need to allow aMule to use 10 kB/s out of about 12 kB/s of upload capacity, which makes web surfing extremely painful (ping google.com gives > 2 secs). I don’t mind sharing my bandwidth, but this is too much.
. An ordinary differential equation (ODE) is similar, except the unknown is a function rather than a number. Such an equation involves derivatives of the function. Here is an example:
, where 
can be any number. Finally, a partial differential equation (PDE) involves a function that has two or more parameters, and includes partial derivatives of this function. For example, the following equation describes waves propagating through a medium:
![f[j] = f[j] + \frac { \partial f } { \partial t } [j] * dt](/latexrender/pictures/3de923578d6c440441bbc686d59bc5bf.png "f[j] = f[j] + \frac { \partial f } { \partial t } [j] * dt")
![f[j] = \frac{f[j-1] + f[j+1]}{2} + \frac { \partial f } { \partial t } [j] \; dt](/latexrender/pictures/08f7b12d31b3ef545da95e66ae44b89a.png "f[j] = \frac{f[j-1] + f[j+1]}{2} + \frac { \partial f } { \partial t } [j] \; dt")
Subscribe by email