Non-Static Spacetime: Is It an Approximation?

[epovo]

I have a problem with static/non-static spacetime. The problem is that the notion of spacetime includes time itself, so how can it change with time?
Imagine an asteroid approaching the Earth-Moon system. The Earth-Moon system is a non-static spacetime, so presumably is giving off gravitational waves. But in principle it should be possible to use GR to calculate the asteroid's trajectory (treating it a test object), in other words, to obtain the equations of the geometry of spacetime of such system. Gravitational waves would be described within that geometry. I suppose that the spacetime geometry around such system is too complicated, so we choose to treat it as a static spacetime and then consider the movement of the two massive bodies as some kind of perturbation. So my question is: is the concept of non-static spacetime a mere approximation to the real thing?

 

[Nugatory]

is the concept of non-static spacetime a mere approximation to the real thing?

I don't know that I'd say a "mere" approximation, although you are correct that there is no real-world situation which is really truly 100% static.

There's nothing special about spacetime here - we routinely ignore effects whose effects are smaller than the accuracy of our measuring devices or the accuracy that we require in our calculations. A pilot calculating the fuel required for a flight from Berlin to Tokyo doesn't bother making corrections for the added weight of a bug squashed into the wheels while he was taxiing out for takeoff.

 

[m4r35n357]

So my question is: is the concept of non-static spacetime a mere approximation to the real thing?

I don't think so, I think the problem you are expressing is that we simply don't have enough closed form non-static solutions to deal with the most interesting cases. Numerical simulation is capable of solving such problems, but (outside academia) setting it up is a nightmare compared to the approximate methods we use in practice.

 

[epovo]

Let me put it another way. Is it not a contradiction to speak of a spacetime geometry that changes with time? That's why I thought that it is an approximation. Surely, a complete geometrical description would include the whole "change with time" within it, gravitational waves and all, wouldn't it?

 

[Nugatory]

Let me put it another way. Is it not a contradiction to speak of a spacetime geometry that changes with time?

Maybe, but that's not what a "non-static spacetime" is. A static spacetime is one that has a timelike Killing vector field.

 

[m4r35n357]

Let me put it another way. Is it not a contradiction to speak of a spacetime geometry that changes with time? That's why I thought that it is an approximation. Surely, a complete geometrical description would include the whole "change with time" within it, gravitational waves and all, wouldn't it?

Numerical relativity is all about the evolution of a dynamic spacetime - the geometry itself - and the matter/energy within it; growing/merging black holes, gravitational waves, the lot. But, as I said, solving Einstein's equations is not straightforward. In fact, you don't even get to do that until you've set up initial/boundary conditions (which is just as hard if not harder). Try searching online for numerical relativity, others can explain better than me. There is even an open source package called the Einstein Toolkit.

 

[PeterDonis]

Is it not a contradiction to speak of a spacetime geometry that changes with time?

No, because "time" has two different meanings in this sentence. The "time" in "spacetime" is a dimension--one of the four dimensions of the spacetime manifold. The "time" in "changes with time" is "time actually measured by a clock traveling on a specific path through spacetime".

Surely, a complete geometrical description would include the whole "change with time" within it, gravitational waves and all, wouldn't it?

Yes. And such a description is what is (sloppily) referred to as "geometry changing with time".

Let me unpack this a little bit to make it clearer what is going on. Consider the spacetime surrounding a spherically symmetric gravitating body whose mass is constant. In this spacetime, one can find a family of observers with the following property: along the worldline of each observer, the metric (the geometry of spacetime) is constant--i.e., it's the same at every point on the worldline. (Notice that the word "time" does not appear anywhere; we are taking the geometric viewpoint where we just have worldlines and different points on them, and we can look at what properties are the same at all points, vs. what properties might change from point to point.) When we say that this spacetime is "static", what we are actually saying is that such a family of observers exists. (These observers are the ones who are "hovering" at a constant altitude above the gravitating body--or standing on its surface, not moving. A more technical way of describing the worldlines of these observers is to say, as Nugatory said, that the spacetime has a timelike Killing vector field, and the worldlines of the observers are integral curves of that vector field.)

By contrast, consider the spacetime that describes the universe as a whole, the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. In this spacetime, it is not possible to find any family of observers along whose worldlines the metric is the same at every point. No matter what timelike worldline you pick in this spacetime, the metric will be different at different points on the worldline. This is what we mean when we say that the geometry of such a spacetime "changes with time": "time" here just means "the parameter we use to distinguish different points on a timelike worldline". We call this parameter "time" because we can measure it using a clock that follows that worldline; but it is a different thing from the "time" dimension of the spacetime, so there is no contradiction involved.

(One technicality should be mentioned here, btw: a spacetime in which a family observers such as I described above exists is actually called "stationary", not static; a "static" spacetime satisfies the additional condition that the spacetime can be foliated by a family of spacelike hypersurfaces that are all orthogonal to the worldlines of the family of observers. Heuristically, a spacetime that is stationary, but not static, can have its central gravitating body rotating, whereas a static spacetime cannot.)