# Can a gravitational wave stretch a volume equally in 3 dimensions?

The wikipedia page for gravitational waves contains this statement: "The area enclosed by the test particles does not change and there is no motion along the direction of propagation. citation needed"

Why citation needed? Is it up for debate?

In my case, clarification needed.

In my own words: It appears that it is not possible for any combination of GWs to stretch space equally in 3 dimensions. This includes any combination of "interfering GWs". For a given volume of space, the volume contained cannot change by any combination of GWs. Any stretching in one dimension must be counterbalanced by a compression in the other dimension(s). This must hold true for every instant of time and every possible sub-division of that volume.

Is this statement true?

Two answers. The first one is that it is not up for debate, in the linearized regime, and the second is yes, it is true. Next it explains this.

FIRST, THE FIRST QUESTION, a citation for whether the stretching or contracting of spacetime can be in the spatial direction it is moving.

Yes a citation would have been helpful in an article so people understand where it comes from. You can find plenty of places where it is proven, and probably in most general relativity books. For sure in Wald. I have not seen a wiki article with the proof. A gravitational wave cannot carry a longitudinal mode, only transverse modes. That is just like the electromagnetic radiating field. In both cases zero mass particles can only be associated with waves that are transverse.

Now, that is true for linearized gravitation waves in vacuum. Just as it is for the electromagnetic field. This has been known and is not an outStanding question. But if you allow nonlinearities, i.e., in a strong gravitational field or simple a very strong gravitational wave, there can be longitudinal modes. None of the gravitational waves we've detected are that strong, and it's not likely we'll see any anytime soon.

There is a nice classification of gravitational field types that was done by Petrov using the Weyl tensor, and independently Pirani, called the Petrov types, and it is a covariant classification (done by how many different principal null directions of the Weyl tensor the resulting spacetime has). Type N spacetimes (or regions of it) only permits transverse waves. Type III was longitudinal waves (again, in the full theory, not the linearized theory), type D has the static gravitational fields, and there are a few others. Type II allows a mix of those in D, III and N.

See the Petrov clasification at https://en.m.wikipedia.org/wiki/Petrov_classification

FOR THE SECOND QUESTION

As for whether a gravity wave can expand the volume of the limited region it goes through, the answer is no, as it seems you know or suspect. The reason is that the curvature of spacetime, determined by the Riemann 4 index tensor, can be defined in terms of the Ricci tensor in combination with the Weyl tensor. Indeed, the Ricci tensor determines how much a volume grows or shrinks, while the Weyl tensor determines the shape, I.e., the distortion of spacetime.

This was already explained in one accepted answer in PSE, at Does curved spacetime change the volume of the space?. The bottom line on that is that the change in volume due to a gravitational wave of a well defined volume determined by a group of particles following geodesics is given by

dV = $[1-1/6R_{ij}x^i x^j + O(x^3)]V_f$

Where $V_f$ is the volume one would see in flat spacetime. This has been known for quite some time.

Now, since the Ricci tensor = 0 in vacuum, to second order the volume does not change. This is not true for very strong fields in non linearized gravitational waves, but it is in the linearized wave regime we will see most of.

So, yes, in the linearized regime gravitational waves are transverse, and do not change the volume. They cause stretching and compressions alternately in two perpendicular directions, and volume is conserved.

• I can (barely) do the math of linearized GWs, but like most mortals I struggle with tensor math. Bobs descriptions are instructive, and just beyond my grasp, so thats good... I'm studying them... Jun 30, 2017 at 0:21
• @Keith I'd recommend just taking the flow of a serious book and it'll get you through the tensor stuff, once you get used to it it's not that hard. A good one, even if someone outdated but they show everything in every detail so you can follow is the Misner, Thorne and Wheeler textbook. Big and not cheap. There's others. The last point about the formula for the volume has been known for a long time, but it's not usually emphasized in text,book, it's sort of specialty knowledge, so I wouldn't sweat that too much. Just go through your own course/book/steps. Jun 30, 2017 at 1:32

Bob Bee has given a correct and rigorous answer. It's hard to tell the OP's level of sophistication, but if that one was too sophisticated for them, a more basic explanation is the following.

Can gravitational waves be longitudinal?

No, and it's fairly easy to see this for a hypothetical purely longitudinal wave. An example of an attempt to create such a metric would be $ds^2=dt^2-[1+(1/10)\sin x]dx^2-dy^2-dz^2$. If you calculate the curvature of this spacetime, it's zero, which means that there is actually no wave present, and this is just a Minkowski spacetime presented in funny coordinates. The idea is that rulers oriented in the $x$ direction are fluctuating in length as a function of $x$, but such a fluctuation can be eliminated simply by recalibrating all the rulers appropriately.

Can gravitational waves cause a volume change?

The Einstein field equations relate a certain type of curvature, which is the non-tidal part, to the presence of matter. A volume change in three dimensions is precisely this type of curvature, and therefore it can't happen in the absence of matter.

• Nice posting Ben Jun 30, 2017 at 1:34