What is time?

I don't discuss here the philosophical problem; only what it means in the context of general relativity, where the notion is not so easily described.

It is commonly asserted that in general relativity there is no absolute simultaneity. On the other hand, it is asserted that, based on the traveling time of light, we see the Sun as it was 8 minutes ago and the Andromeda nebula as it was 2.5 million years ago. This seems to conflict with each other - apparently we have no diffeomorphism invariant way of assigning a relative time to a distant object. Let us take a closer look at the issues involved.

The invariant way of defining present is to say that x and y are present if the two points are in a spacelike relation, and to say y was earlier (or later) than x if y lies in or on the past (or future) light cone. Thus the present is well-defined as the complement of the closed light cone.

Now suppose that you look at the sun. If one is really pedantic, one would have to say that you see the sun in your eye, as a 2D object, and not out there in 3D. But we are accustomed to interpret our sensations in 3D and hence put the sun far away but into the here.

In general relativity, one goes a step further. One thinks in terms of the 4D spacetime manifold and places the sun there. Calculating the length of the geodesic gives a value of 0, so the sun is not in your present. Consideration of the sign of the time component in an arbitrary proper Lorentz frame, one finds that the sun is in your past, as everything you observe.

But the amount of invariant time passed, as measured by the metric, is zero. This looks like a paradox. What happened with the claimed 8 minutes?

The answer is that the metric time is not the right way to measure time. It is the only time available in a Poincare-invariant flat universe, or in a diffeomorphism invariant curved universe. An empty universe where only noninteracting observers move has no notion of simultaneity.

But a matter-filled, homogeneous and isotropic universe generally has one, defined by the rest frame of the galactic fluid with which general relativity models cosmology. Since the fluid breaks Lorentz symmetry (except in very special cases, which are ruled out by experiment) it creates a preferred foliation of spacetime. This foliation gives a well-defined cosmic time, when scaled to make the expansion of the universe uniform. (Actually there are several natural scalings = monotone transformations of the time parameter; see Section 27.9 in Misner/Thorne/Wheeler, so cosmic time without a reference to the scale used is ambiguous.)

This cosmic time figures in all models of cosmology. The values commonly talked about when quoting times for cosmological events, such as the date of the big bang or the time a photon seen now left the Andromeda nebula, refer to this cosmological time.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ