Can you travel faster through time?

If you watch science fiction movies, the most dramatic effects are obtained through some form of time travel. Pick some time in the future, or the past and a fabulous machine or spell swoops you away to that time.

I have always had a problem with this simple approach to time travel. One obvious objection would be this. Remember that the earth is rotating around its axis at 1000 \frac{km}{hr}, revolving around the Sun at 100,000 \frac{km}{hr}, spinning around the center of the Milky Way at 792,000 \frac{km}{hr} and being dragged at 2.1 million \frac{km}{hr} towards the Great Attractor in Leo/Virgo together with the other denizens of the Milky Way! If someone took you away for a few seconds and plonked you back at the same {\bf {SPOT}} in the universe after a few couple of minutes, but several thousands years ago or ahead, you sure would need a space suit. Earth would be several billions of kilometers away, you’d be either totally in empty space or worse, somewhere inside a sun or something. I can’t imagine the Connecticut Yankee in King Arthur’s court carried spare oxygen or a shovel to dig himself out of an asteroid!

OK, so if you built a time machine without an attached space capsule to bring you back to the earth, woe is you. In addition, here is a simple way to sort out the paradoxes of time travel – just put down a rule that you can travel back in time, but you will land so far away that you can’t affect your history! This should surely be possible! Only time will tell.

Anyway, I didn’t really mean to share with you my proposal for how to make time travel possible. I was pondering a peculiarity of Einstein’s theory of relativity that isn’t often made clear in basic courses.

In Einstein’s world view, as modified by his teacher Hermann Minkowski, we live in a four dimensional world, where there are three space axes and one time axis. In this peculiar space, {\bf {Events}} are labeled by (t, \vec x) describing the precise time they occurred and their position (three coordinates give you a vector). In addition, if there are two such events (t_1, \vec x_1) and (t_2, \vec x_2), then the “distance” (t_1-t_2)^2 - (\vec x_1- \vec x_2)^2 is preserved in all reference frames. Why the peculiar minus sign between the time and space pieces? That’s what preserves the speed of light between all reference frames, as Minkowski realized was the key to Einstein’s reformulation of the geometry of space time. It is interesting that Einstein did not think of this initially and decided that Minkowski and other mathematicians were getting their dirty hands on his physical insights and turning them into complicated beasts – he came around very quickly of course, he was Einstein!

To generalize this idea further, physicists invented the idea of a 4-vector. A regular 3-vector (example \vec x) describes, for instance, the position of a particle in space. A 4-vector for this particle would be (t,\vec x) and it describes not just where it is, but when it is. Writing equations of relativity in vector form is useful. We can write them down without reference to one particular observer, without specifying how that observer is traveling.

In notation, the 4-vector for position of a particle is x^{\mu} \equiv (x^0, \vec x) = (t, \vec x).

Before we go any further, it is useful to consider the concept of the “length” of a vector. An ordinary 3-vector has the length |\vec x| = \sqrt{x^2+y^2+z^2}, but in the 4-vector situation, the appropriate length definition has a minus sign, |x^{\mu}| = \sqrt{c^2 t^2 - x^2 - y^2 - z^2}. The relative minus sign again comes from the essential consideration that the speed of light is constant between reference frames. Note that if you look at the 4-vector for an event – it has four independent components – the time and the position in three space.

Next, one usually needs a notion of velocity. Usually we would just write down \frac{d \vec x}{dt} with 3-vectors. However, when we differentiate with respect to t, we are using coordinates particular to one observer. A better quantity to use it one that all observers would agree on – the time measured by a clock traveling with the particle. This time is called the “proper time” and it is denoted by \tau. So the relativistic 4-velocity is defined as V^{\mu} = \frac{d x^{\mu}}{d \tau}. In terms of coordinates used by an observer standing in a laboratory, watching the particle, this is

V^{\mu} \equiv (V^0,\vec V) = (\frac{1}{\sqrt{1-v^2/c^2}}, \frac{\vec v}{\sqrt{1-v^2/c^2}}).

If you haven’t seen this formula before, you have to take it from me – it is a fairly elementary exercise to derive.

The peculiar thing about this formula is that if you look at the four components of the velocity 4-vector, there are only three independent components. Given the last three, you can compute the first one exactly.

In the usual way that this is described, people say that the magnitude of this vector is (V^0)^2-(\vec V)^2=1 as you can quickly check.

But it is peculiar. After all x^{\mu}, the position, does have four independent components. Why does the velocity vector V^{\mu} only have three independent components. Those three are the velocities along the three spatial directions. What about the velocity in the “time” direction?

Aha! That is \frac{dt}{dt}=1. By definition, or rather, by construction, you travel along time at 1 day per day, or 1 year per year or whatever unit you prefer. The way the theory of relativity is constructed, it is {\bf {incompatible}} with any other rate of travel. {\bf {You \: \: cannot \: \: travel \: \: faster \: \: or \: \: slower, \: \: or \: \: even \: \: backwards, \: \: in \: \: time \: \: without \: \: violating \: \: the \: \: classical \: \: theory \: \: of \: \: relativity}}.

The only relativistically correct way one can traverse the time axis slower is to rotate the time axis – that’s what happens when then observer sitting in a laboratory hops onto a speeding vehicle or spaceship, i.e., performs a Lorentz transformation upon him/herself. That’s what produces the effects of time dilation.

Your only consolation is that virtual quantum particles can violate relativity for short periods of time inversely proportional to their energy. However, they are just that – virtual particles. And you cannot create them for communication.

Oh, well!

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