Conceptualizing spacetime as static or evolving

Question

I've been using this model to wrap my head around the concept of spacetime: https://www.youtube.com/watch?v=wrwgIjBUYVc

In the video you'll see an animation of an apparently dynamic and warping 3-space that is supposed to be representative of spacetime. But earlier in the video, he stacks layers of 2-space for a flatland model of spacetime. In that view, the spacetime manifold appears static, with time axes extruding into the 3rd spatial dimension (of the abstract manifold).

Is it better to think of spacetime as static or dynamic? In the static model, we think of a bunch of geodesics cast onto a static background of curved spacetime. In the dynamic model, we think of bodies being enmeshed in and co-moving with space-time, with a relative component fighting against the "river" of flowing spacetime. Thus, a satellite orbiting a massive body is conceived of as not following a geodesic in a static manifold, but constantly getting sucked in by a current of deforming spacetime. If a planet is seen as a sink of space, what happens to the space when it reaches the center of a planet? Does it get annihilated?

Which is the more correct interpretation?

Answer 1

In the static model, we think of a bunch of geodesics cast onto a static background of curved spacetime.

Contrasted to your "dynamic model", your "static model" is more faithful to what spacetime objectively is. It's a 4-dimensional manifold. It can't evolve with time -- it already encompasses time!

However, the idea of "geodesics cast onto a static background" can be also misleading. That background is completely determined by the energy content, i.e. by those same geodesics.

In the dynamic model, we think of bodies being enmeshed in and co-moving with space-time, with a relative component fighting against the "river" of flowing spacetime.

The problem with this kind of picture is that it is always coordinate-dependent. Within the context of general relativity, the only coordinate-independent property of a point in spacetime is its (tensor) curvature. There is no objective sense in which it can exhibit motion.

The "river" interpretation has particular problems, including the one you have raised. There is not really any concern about space being "annihilated" at the center of a planet. Also, it is equally valid to construct coordinates in which the "river" flows outward, away from gravity sources, rather than inward. See the answers to this question for further discussion of the river interpretation.

Answer 2

In general, spacetime is dynamic. Dynamic just means that the geometry of spacetime (i.e. the metric tensor, a mathematical object which encodes the information about the geometry) will depend on some notion of time.

It's important to emphasize that you cannot tread space and time differently in relativity. This is because, due to the nature of Lorentz transformations, what one observer calls "time" will be seen as "space" by another.

You may find a specific coordinate system in which spacetime appears static, meaning the metric tensor will not change with time as measured by a clock in that specific coordinate system. But once you go into another coordinate frame, the geometry may look "dynamic" again.

As a simple example, the curvature of spacetime around the earth looks static to an observer on the surface, but to an observer flying away from earth with some velocity, it won't.

Thus it is true that whenever you have a mass moving through spacetime, the geometry will change. But often, we can approximate spacetime as static (in a specific coordinate system). If you consider the movement of the earth around the sun, the curvature created by the earth will be negligable compared to the curvature created by the sun, simply because it has a lot more mass.

If a planet is seen as a sink of space, what happens to the space when it reaches the center of a planet? Does it get annihilated?

I'm curious as to why you would think that something spectacular happens at the center. A planet is not a black hole where you have infinite density, and therefore infinite curvature at the center. The planet has finite density everywhere, so you get finite curvature. It simply leaves a little "dent" in spacetime, if you will. Nothing gets annihilated.