# Can a gravitational wave (GW) propagate in a flat universe?

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• grauitate
In summary: This is a significant point, and I would like to quote from a textbook on the subject:"The concept of a gravitational field is difficult to define unambiguously. It is not clear what the physical quantity that should be measured to determine the existence of a gravitational field. In general, it might be thought that the presence of a gravitational field should be detectable as a change in the motion of particles around a gravitational center. However, this is not the case. The general theory of relativity predicts the existence of gravitational fields even in the absence of any masses or masses in equilibrium. In other words, the gravitational field is the result of the gravitational interaction between masses.""All GR textbooks are quite clear on the fact that
grauitate
As the universe expands and is per definition gravitationally decoupled on long distances and the overall metric therefore is "flat" and apparently no gravitational background exists, the question in some discussion arose:
Can GWs propagate in in a gravitational empty space at all?
If not, and as an outcome of the recent detection of GWs, I then suppose that the universe cannot be flat.
There should be a kind of an overall and isotropic gravitational bias (of course only positive, as negative gravitation does not exist) in which e.g. the grav. oscillating "glitch" of heavy bodies orbiting each other can propagate (by the speed of light).

Your definition of "empty" - flat metric - precludes gravitational waves. So your question boils down to "are there gravitational waves in a region definied not to have gravitational waves".

Space is flat. Not spacetime. Also, gravitational waves can propagate in a flat Minkowski background. The Einstein eqns. only constrain the metric. They don't determine it completely. (This assumes more than 2 spatial dimensions)

haushofer said:
gravitational waves can propagate in a flat Minkowski background.

To be more precise, gravitational waves can propagate in a spacetime where the stress-energy tensor is zero everywhere. But the term "flat Minkowski background" can be misleading: the spacetime is not flat, it's curved. It just happens that the curved metric can be written as the flat Minkowski metric plus small correction terms.

haushofer
Just reading between the lines, I wonder if the O.P. is confusing a 'flat' metric which can be perturbed with the 'absence' of any field where there would be nothing to perturb.

I would reply to them as follows:
"So your question boils down to "are there gravitational waves in a region definied not to have gravitational waves".
Of course, a GW is emitted in a local system of appreciable grav. background and curvature of metric, e. g. in a galaxy or in the realm of a galaxy cluster with a characteristic size of 1...10 Million Ly - but as the source of the first GW detection was localized at z ≈ 0,09, i.e. in approx. more than 1 Billion Ly this is far out of grav. interaction but within the full range of space expansion. Here space is not curved by any gravitation because the accelerated expansion takes over the equivalent role for stabilisation. Mach's principle is abandoned here also. By observing the far distant objects we see them only by the time distorting effects of Special Relativity (redshift, time dilation of SNe and other processes)
"Space is flat. Not spacetime"
Yes.
"Also, gravitational waves can propagate in a flat Minkowski background"
I'm not sure
"To be more precise, gravitational waves can propagate in a spacetime where the stress-energy tensor is zero everywhere".

I may answer with a citation from Wikipedia "Stress energy tensor": "The stress–energy tensor is the source of the gravitational field in the Einstein field equations of general relativity"
So, if the stress-energy tensor is zero everywhere, no gravitational field exists which, as I already said, could serve as a positive base or bias for the propagation of the fluctuations or perturbations in it caused by the GWs.
"Just reading between the lines, I wonder if the O.P. is confusing a 'flat' metric which can be perturbed with the 'absence' of any field where there would be nothing to perturb"
I think this is the point - my understanding is that the accelerated expansion of space which is effective on large distances compensates completely for any gravitational field, so gravitation is completely canceled out and "there would be nothing to perturb" any more.

grauitate said:
I may answer with a citation from Wikipedia

Which is not an acceptable source. You need to be looking at textbooks and peer-reviewed papers. All GR textbooks are quite clear on the fact that vacuum solutions (i.e., solutions with zero stress-energy tensor everywhere) of the Einstein Field Equation exist that describe propagating gravitational waves.

grauitate said:
if the stress-energy tensor is zero everywhere, no gravitational field exists

This is not correct, any more than it is correct that no electromagnetic field can exist where there are no charges or currents. This is what comes of trying to use Wikipedia as a source. (This is even if we set aside issues with the term "gravitational field" in the first place--see below for more on that.)

Even if we leave out gravitational waves, there are vacuum solutions of the Einstein Field Equation that describe black holes--static objects made of pure spacetime curvature that have "gravitational fields" in the same sense as ordinary planets or stars. So the Wikipedia statement is obviously false.

grauitate said:
my understanding is that the accelerated expansion of space which is effective on large distances compensates completely for any gravitational field

I don't know where you are getting this understanding from. Can you give a reference?

One obvious problem is that the concept of "gravitational field" doesn't even make sense on cosmological scales in an expanding universe.

PeterDonis said:
To be more precise, gravitational waves can propagate in a spacetime where the stress-energy tensor is zero everywhere. But the term "flat Minkowski background" can be misleading: the spacetime is not flat, it's curved. It just happens that the curved metric can be written as the flat Minkowski metric plus small correction terms.
Yes, you're right, that can be Very misleading indeed, never thought about that before.

PeterDonis said:
Which is not an acceptable source. You need to be looking at textbooks and peer-reviewed papers. All GR textbooks are quite clear on the fact that vacuum solutions (i.e., solutions with zero stress-energy tensor everywhere) of the Einstein Field Equation exist that describe propagating gravitational waves.

Sorry, maybe I'm a little oldfashioned, but under historical respect Einstein in 1916 and 1918 developed his GW-equations on the base of his stationary universe, which is curved and closed and where Mach's principle (MP) and the strong eqivalence principle (EP) was valid and last not least a isotropic (scalar) mean constant grav. field was existing everywhere. Here is an interesting comment of Eddington in 1922 "The Propagation of Gravitational Waves"
http://rspa.royalsocietypublishing.org/content/royprsa/102/716/268.full.pdf
which starts with the sentence: "The problem of the propagation of disturbances of the gravitational field was investigated by Einstein in 1916 and 1918."
So, pure logically spoken, where there is no grav. field no propagation of GWs should be possible. This should be no problem within the vicinity of a galaxy or a galaxy cluster, but far out in the realm of the accelerated expansion of space, MP and EP are abandoned and with them a mean isotropic grav. field of the universe.
So, from my favorite textbook: Max Born, Die Relativitätstheorie Einsteins, Springer Vlg. Berlin, Heidelberg, NewYork, 1964,1969, 2001, 2003,
the 10 metric coefficients g11,...g34 describe the generalized theorem of the fourdimensional world.
Now if there is no gravitational field at an origin O it applies (page 292, equat. 99)
g11 = g22 = g33 = 1, g44 = -c2
and
g12 = g13 = g14 = g23 = g24 = g34 = 0
Within this metric no GW can be derived.

So at least you have to introduce a grav. field, but only a weak one, which I'm free to cite here (page 358,359)
g11 = 1 + h11, g22 = 1 + h22, g33 = 1 + h33,
g44 = -c2 + h44
and correspondingly
g12 = h12, g13 = h13, g14 = h14
g23 = h23, g24 = h24, g34 = h34
Now taking bij as the twofold derivative of the coefficient hij for the time and fij as the sum of its twofold derivatives for each of the three space-coordinates, so for each coeffizient hij a wave equation can be written as
bij = c2 fij
So the prerequisite of a GW propagation is a gravitational field and now detecting GWs from distances as far as more than a Billion Lys would implicate a curved universe and not a flat one. !(?)

grauitate said:
under historical respect Einstein in 1916 and 1918 developed his GW-equations on the base of his stationary universe

The Einstein static universe is not a gravitational wave solution. The investigations Eddington referred to in the quote you give were, AFAIK, investigations of the linearized approximation to GR. They had nothing to do with the Einstein static universe.

grauitate said:
where there is no grav. field no propagation of GWs should be possible

The term "gravitational field" is vague. If you actually look at the math, you will see, as I said before, that there are vacuum solutions of the Einstein Field Equations that describe propagating gravitational waves. So the answer to your question in the OP is yes. The sources you are looking at do not contradict that--and anyway, the basis for my statement is not those sources but decades of work since then in which the possible solutions of the EFE have been thoroughly investigated. So we today know a lot more than Einstein and Eddington did, and that up to date knowledge is what you should be looking at.

grauitate said:
the prerequisite of a GW propagation is a gravitational field and now detecting GWs from distances as far as more than a Billion Lys would implicate a curved universe

Yes, that's correct. The vacuum solutions I referred to, that describe propagating gravitational waves, are curved spacetimes, not flat Minkowski spacetime. As has already been pointed out, you should not confuse flat space with flat spacetime.

And with that, this thread is closed since the OP question has been answered.

## 1. Can gravitational waves propagate in a flat universe?

Yes, gravitational waves can propagate in a flat universe. In fact, the theory of general relativity predicts that gravitational waves can travel through any type of universe, whether it is flat, curved, or expanding.

## 2. How do gravitational waves travel in a flat universe?

Gravitational waves travel through a flat universe at the speed of light, just like electromagnetic waves. However, unlike electromagnetic waves which require a medium to travel through, gravitational waves can propagate through the vacuum of space.

## 3. Can gravitational waves affect the expansion of a flat universe?

Yes, gravitational waves can affect the expansion of a flat universe. In fact, in the early stages of the universe, gravitational waves played a significant role in driving the rapid expansion of the universe known as inflation.

## 4. Are gravitational waves affected by the presence of matter in a flat universe?

Yes, gravitational waves can be affected by the presence of matter in a flat universe. According to general relativity, matter can create ripples in the fabric of space-time, which can in turn affect the propagation of gravitational waves.

## 5. How can we detect gravitational waves in a flat universe?

Gravitational waves can be detected in a flat universe using highly sensitive instruments known as interferometers. These instruments measure tiny changes in the distance between two objects caused by passing gravitational waves. The most famous interferometer used for detecting gravitational waves is the Laser Interferometer Gravitational-Wave Observatory (LIGO).

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