# How do gravitational waves work without internal tension?

One implication of general relativity is the concept of gravitational waves or gravitational radiation, ripples in spacetime thought to travel at speeds close to the speed of light. As far as I have researched, there is no direct evidence supporting their existence, but I have read many examples of indirect evidence.

My question is regarding the nature of these proposed waves. I understand the motivation behind the idea of a gravitational wave, but I do not understand the reasoning behind how said wave is created. For instance, how is it possible for a wave to be created in spacetime, unless we make the assumption that spacetime (like a fabric) has some form of internal tension? When I attempt to visualize waves I tend to think back to sound waves and ocean waves...but these waves rely on internal forces between molecules in the medium to create the ripples. Do similar mechanics take place in a gravitational wave or is there some other explanation? Thank you for any help with this question.

The fact of a wave doesn't needfully mean there's tension or elasticity in the sense of those phenomena in an acoustic medium. General Relativity describes how spacetime's geometry depends on the distribution of energy within it[1]. But GTR does not tell us anything about the microscopic "machinery" of spacetime that gives rise to this behavior: tension driven or otherwise. That will hopefully be the main results of a future theory of quantum gravity. The form of the Einstein Field Equations was reasoned purely from the equivalence principle (which suggested a manifold as the geometrical object), properties of tensors (whose language was used to make the the theory independent of co-ordinates and background) and analogy with the Poisson equation: see Eduardo Guerras Valera's most wonderful answer here, which itself is a great summary of Einstein's lucid little book The Meaning of Relativity, both recommended reading.

Even outside GTR, "wave" doesn't needfully imply "tension" even though it's absolutely true that tension is one way wherein waves can arise. I guess it depends on one's definition of a wave, but for me something that fulfils D'Alembert's Wave Equation pretty much is as "wavy" as one can get. Indeed the disturbance to the metric tensor in the Weak Field Einstein Equations fulfils precisely this equation (see also §8.3 "Einstein's Equations for Weak Gravitational Fields" in Bernard Schutz, "A First Course in General Relativity"). As you will likely know, the solution to the one dimensional D'Alembert wave equation is an arbitrary linear superposition of any set of functions of the form form $f(z\pm c\,t)$, and such functions in turn imply the D'Alembert Wave Equation. But a microscopic army of invisible beings playing whisper down the lane can beget macroscopic wave motion described by the D'Alembert equation, and no tension is involved. Likewise, at a microscopic level, sound does not really involve tension, just collisions: it's pretty near to the whisper down the lane game.

Although I'm not greatly fond of the "fabric" analogy, there's a grain of truth in it insofar that "empty spacetime" is not a void. It is a medium in the sense that it has definite, measurable properties that can be different in the neighborhoods of different events in spacetime: witness the non-flat metric for a Schwarzschild black hole with a variable curvature tensor: this means that the "empty" spacetime regions around the black hole (let's keep our discussion outside the event horizon) have different geometries, deviating more severely from Euclid's parallel postulate the nearer the horizon on gets. Modern physics thinks of empty space as being made of quantum fields: an "empty region" is simply one where the quantum fields are all in their ground state, and there is no need of a "void" further to these fields. The main thing, though, that is different for this medium as opposed to what people wontedly think of "mediums" is that so far all experiments support Galileo's relativity principle, that there is no experiment that one can do or any measurement that one can make within one's own laboratory that would detect one's motion relative to any "absolute frame". So almost all of the 19th century aether theories are ruled out by experiment[2], because their postulate behaviors would give us an easy way to detect one's motion relative to the medium: e.g. the acoustic wave equation changes its form depending on one's relative motion to the acoustic medium.

[1]. It's true that this distribution is the "Stress-Energy" tensor, but this "stress" refers to fluxes of momentum that arise within spacetime owing to the interaction between non-ground-state "stuff" within spacetime in the same way that stress in a beam (the thing a civil engineer works with, not a light beam!) measures the momentum flux across planes owing to the interaction between different molecules making the beam up. Gravitational waves can propagate in space where the stress energy tensor is nought. John Duffield's answer cites the cosmological constant as a kind of "stress", which is not incorrect. However, gravitational waves are still a feature of the Einstein field equations even if the CC is nought: so far we have no reason to believe - either theoretical or experimental - that a nonzero cosmological constant is necessary for gravitational waves.

[2]. But see the Lorentz Aether Theory that yielded identical predictions to special relativity. See also ACuriousMind's description here

One simplification of what gravitational waves are and how they're created, can be understood by visualizing a massive body and its gravitational field when stationary and what happens when the body accelerates.

Imagine what happens if such a body were to start accelerate to the right for a while, and then stop. Just like the time-delay with light, it will take time for distant objects to notice the motion / change in position of a distant gravitational source. In one sense, the motion has created a ripple effect for distant observers - on the right, observers will notice that around the time they can see this body move, the gravitational field at their location has been increased in strength; in other words, a ripple in the curvature propagated to the right that resulted in slightly more curvature for the observers on the right than before. A similar effect would occur for observers on the left with a weakening strength.

These 'ripples' so far do not need a medium for the effect to be understood. The 'medium' here, would be to have an observer at every point in between. That is, if you could observe this effect from everywhere and collect the results, you would see a propagating wave in the data despite awareness of any medium.

Any phenomena that involves significant energy densities and accelerations, like stars, black holes, galaxies or even events near the epoch of the big bang etc, would be capable of producing significant gravitational waves.

I realize this doesn't directly answer whether or not gravitational waves are due to spacetime tension or similar. Strictly speaking it's not right to think of it as tension...

If we were dealing with an electric charge and its electric field... The change in electric field due to motion, would induce a changing magnetic field (while the motion isn't constant), which in turn induces a changing electric field etc - propagation in a sense. You can think of gravitational waves as having this sort of interplay, but instead of electric charge you have mass 'charge', so the interplay is between mass-energy and curvature, each being the cause of the other.

As consequence, gravitational waves do carry mass-energy and exhibit their own gravitational field, odd as that may sound.

• Thanks @Xeren, that helped. I figured it would be similar to EM waves but I didn't quite understand how. So, if I am understanding you correctly, I should think of a gravitational wave as the product of fluctuations in mass-energy and space-time as opposed to thinking of spacetime as an elastic medium? Aug 6, 2015 at 3:36
• More or less, yes. It's still an oversimplification of the picture I must stress though. Anywhere there's space-time curvature, there is mass-energy and vice versa. Aug 6, 2015 at 3:42

One should clarify that there exist different frameworks for answering this question.

The classical and the quantum mechanical. The reply by Xeren addresses the classical framework, i.e where , as with the E and B fields of classical electromagnetic radiation there exists the G field , a tensor field, and in a similar way to electromagnetism builds the gravitational wave.

The holy grail of current theoretical physics is the quantization of gravity within a unified filed theory. In this framework the gravitational waves will be built up by gravitons again similar to the way the classical em wave is built up by photons.

In both frameworks it is the electromagnetic wave that is the correct analogue as no medium is required for the propagation.

How do gravitational waves work without internal tension?

There is an internal tension. Google on negative pressure cosmological constant. Tension is negative pressure.

One implication of general relativity is the concept of gravitational waves or gravitational radiation, ripples in spacetime thought to travel at speeds close to the speed of light. As far as I have researched, there is no direct evidence supporting their existence, but I have read many examples of indirect evidence.

Yes, see Taylor and Hulse. And don't forget that the E=hf photon has a non-zero "active gravitational mass". So in addition to being an electromagnetic wave, you could claim that it was in some respects a gravitational wave too.

My question is regarding the nature of these proposed waves. I understand the motivation behind the idea of a gravitational wave, but I do not understand the reasoning behind how said wave is created. For instance, how is it possible for a wave to be created in spacetime, unless we make the assumption that spacetime (like a fabric) has some form of internal tension?

That's how it is. Have a look at the LIGO website: "Space-time can be thought of as a 'fabric' defined by the measuring of distances by rulers and the measuring of time by clocks. The presence of large amounts of mass or energy distorts space-time - in essence causing the fabric to 'warp' - and we observe this as gravity".

Do similar mechanics take place in a gravitational wave?

Yes. Take a look at the stress-energy-momentum tensor which "describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics". Note the shear stress term, which tells you that we're dealing with something like continuum mechanics, and follow the link to the Cauchy stress tensor which completely defines "the state of stress at a point inside a material in the deformed placement or configuration". A gravitational field is "stressed space", and a gravitational wave is where this is propagating.

Note that it's a popscience myth that EM waves require no medium. Especially the claim that the E wave creates the M wave which creates the E wave. See electromagnetic radiation on Wikipedia and note this: "the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time”. If it was an ocean wave and you were in a canoe, E denote the slope of your canoe and B denotes the rate of change of slope. There's aren't two waves present, just one, and it's an electromagnetic wave. Another popscience myth is that space isn't a medium. See this, and this, and make sure you read the Robert B Laughlin quote here: "It is ironic that Einstein's most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed".