How does 4D spaceTIME itself evolve over time? this seems like 5 dimensions: spacetime-time

Question

I think my question is largely a duplicate of this one. But I'm not sure.

Isn't the following quote by John Archibald Wheeler a bit circular, almost a chicken-and-egg situation?

Spacetime tells matter how to move; matter tells spacetime how to curve.

Specifically, I am envisioning a situation where mass moves through time into the future. As it does this, I also envision it changing its location in space, let's assume traveling with constant velocity. I suppose as it does this, it creates waves in spacetime so that it can communicate the fact that its location and thus gravitational effects have changed to the rest of the universe. This wave will propagate with time, deforming 4D spacetime. But on some level, it does not make any sense for the shape of 4D spacetime itself to change and evolve with time. This would mean there are two dimensions of time: the first that is part of 4D spacetime, and a second across which that 4D space itself evolves.

I think my question is probably the same as this one because it asks about when/why we treat spacetime as dynamic versus static. In some situations, it seems like we treat 4D spacetime as a static, unchanging object through which various objects have fixed and timeless worldlines. Other times, it seems like the spacetime itself adjusts dynamically as moving mass warps it and causes gravitational waves. Without thinking about it too hard, this seems contradictory. How can I have a fixed and timeless worldline in 4D spacetime if the space itself is changing with time?

One partial possible answer is the comment from this answer written by ohneVal which discusses how objects that experience movement in space due to gravitational waves experience exactly that—movement in space, and only space. But I'm not sure how helpful this is because ultimately it seems like the objects will still be moving in some way through spacetime, and that this spacetime must evolve over time because if it didn't, then how else would its curvature ever change if not through time passing?

Another possible partial answer is this, which says spacetime itself can have an acceleration vector but not a velocity vector, but not much is said about what this means.

Another interesting idea is in this and this answer, which state that is wrong to think of 4D spacetime as moving:

The river picture leads people to wonder why black holes don't just suck up everything around them, and how the space that flows into them is replenished. Those would be reasonable questions if the river picture made sense, but it just doesn't.

Truth be told, I am curious what the source of spacetime is that replenishes it as it gets sucked up into the black hole; for rivers, we have snowmelt and rain.

Finally, the duplicate question does have some partial answers, for example:

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!

but I find this a bit unsatisfactory in the sense that simply stating spacetime does not evolve with time does not really explain how its curvature is supposed to change if not through or with time. The other answer seems to misinterpret the mental model of a "static" spacetime being envisioned by the question asker.

Notes: maybe relevant previous posts: static spacetime, visualizing spacetime not too relevant previous posts: i think this is asking if time is even a dimension at all, multiple time dimensions do exist in string theory, how to visualize multiple time dimensions other links worth looking at: reddit

Answer 1

Spacetime doesn’t evolve over time, but our position in it does

I hope I am understanding your question correctly.

The problem statement: how can spacetime change over time if time is a dimension?

The answer: how do things change over space?

• If the [energy density] at this [position] is $x$ and its derivative with respect to [position] is $y$ then you can integrate forward to obtain the [energy density] at [other positions].
• If the [curvature] at this [time] is $x$ and its derivative with respect to [time] is $y$ then you can integrate forward to obtain the [curvature] at [other times].

You treat spacetime just like you would anything else, except time is an axis instead of a parameter.

What can’t happen is the curvature at a given event $X=(t,x,y,z)$ changing. The curvature (and in fact the anything) at a given event is constant and does not “change with time” because that is nonsensical. At a given event, there is a given curvature.

The curvature at future events is (in pure general relativity) known from the beginning of the Universe. The spacetime metric itself is “constant” in the sense that there is a single metric that describes physical reality. The metric itself does not change with time, i.e. the curvature at any given event remains constant because there is nothing with respect to which it can change. But the curvature on a worldline, i.e. the perceived curvature of an individual, can very well change because the position of the “present” on that curve changes with respect to the $t$-coordinate (or doesn’t).

Answer 2

Spacetime can certainly evolve with time. That does not mean there is a new dimension or variable to deal with. It simply means the spacetime is a non-constant function.

A simple example: gravity near Earth's surface is constant, $g$. So calculation of an object's velocity or trajectory simply involves integrating a constant force:

$$v(t) = \int g~dt$$ $$= g\int dt$$

etc. But if the object is, say falling while attached to a spring, the net force on the object is itself a function of time, and of the trajectory:

$$v(t) = \int ~g(t)~dt$$

So the calculation is more complicated, but there are no extra variables.

In Relativity we describe the structure of spacetime using a function $g_{\alpha \beta}$ called the metric. This function determines how distances and times relate to one another, and is influenced by the mass and energy in the vicinity. And the metric itself can be, and often is, itself a function of time and space $g_{\alpha \beta}(t,x,y,z)$.

Answer 3

There could be many viewpoints on spacetime within general relativity, some are more suitable for specific problems than others. Usually we view the spacetime of GR as being solution of self-consistent system of Einstein equations representing joint evolution of both geometry and matter fields.

… I am envisioning a situation where mass moves through time into the future. As it does this, I also envision it changing its location in space, let's assume traveling with constant velocity. I suppose as it does this, it creates waves in spacetime so that it can communicate the fact that its location and thus gravitational effects have changed to the rest of the universe. This wave will propagate with time, deforming 4D spacetime. …

With some tweaking and clarifications the viewpoint suggested by OP is a valid way to think about certain general relativistic problems. The key is to see this as the first steps of perturbative contruction of (four-dimensional) spacetime representing joint and self-consistent evolution of background spacetime together with the additional source (in this case a moving mass). One widely used version of such perturbative approach is post-Newtonian expansion.

This would mean there are two dimensions of time: the first that is part of 4D spacetime, and a second across which that 4D space itself evolves.

The “time” can be the same, instead there are multiple (at least two) spacetimes, the first one (“background spacetime”) is a spacetime without the effects of mass, the second is the spacetime perturbed by the passing “wave”. Since the mass exists at all times (from infinitely distant past to distant future) generally the “wave” has time to reach (and thus modify the geometry there) every point of the original spacetime. Note, that coordinates of that second spacetime (including the separation into “space” and “time” parts) could be the same, it is the gravitational field (metric, curvature tensor and other geometric data) that changes. The motion of the mass on this second spacetime would generally be different than on the first, so there would be perturbation to the motion of the mass which in turn would mean second order perturbations to the “wave” which would mean that there should be a third spacetime representing “perturbation of perturbation” of spacetime. And so forth. So, rather than a having “second time”, we can think instead of an infinite sequence of spacetimes (and sequence of motions of the mass on those spacetimes) labeled by an integer parameter: order of approximation. Zeroth order is the background spacetime without the mass, while further spacetimes of the sequence represent spacetime with gravitational effects of the mass incorporated up to a certain order. We can hope that the higher the order the smaller the corrections would be. Generally, such perturbatively constructed spacetime at a finite order of approximation is not a solution of Einstein equations, since at each order we neglect all corrections of higher orders. Fully consistent solution would emerge only as a limit as order of approximation goes to infinity. Although in certain (very special) circumstances approximate solutions at a certain order can become exact and corrections of higher order vanish.

Such perturbative viewpoint on spacetime can be seen as stemming from dichotomy matter—spacetime in the Wheeler's quote, we have the sequence of actions: “move — curve — move — curve …” with each action representing more and more precise approximation to the inherently self-consistent reality.

An aside: Calling perturbations discussed above a “wave” is technically incorrect. Though such perturbations can be seen as solutions of “wave equation” they do not (generally) represent freely propagating gravitational field (gravitational wave) but rather gravitational field bound to a specific source or a combination of bound and propagating field. Just like in electrodynamics a Coulomb field is a solution of wave equation with certain source but is not a wave per se.

Example: A simplest background is a Minkowski spacetime with the metric $$ g^{(0)}_{\mu\nu} dx^{\mu}dx^\nu=-dt^2 +dr^2+r^2 (d\theta^2+\sin \theta d\phi^2), $$ (we use “mostly plus” signature, spherical coordinates and units with $c=1$). A simplest motion of a point-like mass $M$ in this background spacetime is a mass remaining at rest at spatial origin $r=0$.

To obtain the pertubation of spacetime metric for this “motion” we impose a suitable gauge on metric perturbation $h_{\mu\nu}$. The linearized Einstein equations then become a simple wave equation on Minkowski background: $$ \Box h_{\mu\nu}= j_{\mu\nu}.$$ Where $j$ is tensor current corresponding to a point-like source localized at spatial origin. Up to a tensor structure this solution would be a Coulomb field: $$ h_{\mu\nu}dx^\mu dx^\nu=\frac{2GM}{r}(dt-dr)\cdot(dt-dr).$$

Thus the metric of perturbed spacetime at first order of approximation is a sum of Minkowski metric and this perturbation: $$ g^{(1)}_{\mu\nu} dx^\mu dx^{\nu}=(g^{(0)}_{\mu\nu}+h_{\mu\nu}) dx^\mu dx^{\nu}=-dt^2+dr^2+r^2d\Omega^2+\frac{2GM}{r}(dt-dr)\cdot(dt-dr).$$ This is an exact Schwarzschild metric written in Kerr-Schild coordinates, meaning that this is a situation with absent higher order corrections (this occurs due to specially chosen gauge for metric perturbation and the simple law of motion for a single mass).