# How do gravitational waves propagate?

How do gravitational waves work like water waves? Spacetime isn't supposed to be a surface, it is solid, so how do gravitational waves propagate through it like transverse waves on the surface of water since, as far as I know, there is no surface of spacetime? So how do gravitational waves travel through a solid medium? And also can they interfere with each other? Please mention any relevant equations and sorry for the dumb question, I can't find any information online or in books

• see this answer physics.stackexchange.com/q/245214 . Keep in mind that waves are periodic changes , a medium is not necessary. They are modeled by differential equations that have sine and cosine solutions. Jun 11 at 14:25
• I think you mean that a surface is two-dimensional and that spacetime is three-dimensional (actually it's four-dimensional) and has a volume. Spacetime certainly isn't solid! And any n dimensions can be thought of as the surface of n+1 dimensions, curved in those n+1 dimensions. The circumference of a circle is the 1-dimensional "surface" of a 2-dimensional "solid", albeit curved in 2 dimensions. a 4-dimensional "surface" can be curved around a 5-dimensional "solid", etc. Jun 12 at 0:51
• Re "...through a solid medium": See Michaelson-Morley experiment. I couldn't find a canonical question about ether here, but it must exist. Jun 12 at 10:15
• It is not a stupid question, but it's hard to believe that you cannot find info on this online or in books. Already Wikipedia explains it quite clearly with pictures, videos and equations not to mention the large number of simple-level explanations on YouTube. I wonder what books did you try? Jun 12 at 15:59

There are a few misconceptions in the question. I'll try to address them in order of abstraction.

### Waves don't propagate only on surfaces

I believe you are thinking of waves as surface waves, such as the ones you see when you throw a pebble on a lake. However, those aren't the only sort of waves. There are also waves that travel through space. Sound is an example: sound waves are generated by your TV speakers and detected at your eardrums, still there is no surface needed for the propagation. It can propagate through volumes as well. There is no necessity to constrain waves to two dimensional surfaces.

In the comments, Peter Mortensen also mentioned how waves are generated at earthquakes, or how animals and human-made sonars use waves to locate themselves in space. Those are other examples of waves moving through space without the need of a surface. The propagation of light through space can also be added to the list (and, while sound is a longitudinal wave, light is a transverse wave).

### Spacetime is not solid

Spacetime is not fixed nor solid. You seem to be thinking of Galilean spacetime, which is indeed fixed. However, gravitational waves are a prediction of General Relativity, which is the best description we have today of gravitational physics and involves describing spacetime as a dynamic entity. Spacetime isn't solid, but rather it evolves and curves. Gravitational waves are disturbances in the spacetime structure that travel across it at the speed of light. This is a bit analogue to surface waves on a lake: the surface tension on the water makes it so that if you raise a tiny amount of water, the tiny amounts next to it raise a bit as well. When you let it go, they start moving up and down as the pieces next to them push and pull. Spacetime is way more abstract, but the essential idea is similar: a little region of spacetime is bent and this influences the regions nearby; as it oscillates up and down a pulse propagates. These ripples are generated by some strong gravitational fields, such as those around two colliding black holes. The black holes orbiting around each other disturb spacetime, and these disturbances propagate via this "push-and-pull-like" mechanism, until they can, for example, reach Earth and be detected by LIGO and Virgo.

Notice that there is no need of a surface to vibrate. We can also make an analogy with sound waves moving through the air: the pressure on a region makes the surrounding regions move, and consequently changes the pressure in those regions. The effect then propagates, as different regions get larger or smaller pressures and, as a consequence, push on each other. This analogy is a bit more realistic in the sense that gravitational waves oscillate along the directions of space on which they travel, not on some abstract surface.

Disclaimer: Spacetime is not a material medium and the "push-and-pull-like" should be taken with a grain of salt. It is intended to provide intuition, not to be taken literally.

• Sounds like this is describing a wave on an extra-dimensional surface. Jun 12 at 1:39
• @user19087 I'm not sure I understand what you mean Jun 12 at 1:58
• A higher-dimensional surface would encompass a lower dimensional volume, such as space-time, right? This answer makes it seem like there is a higher-dimensional medium (made of what?) upon whose surface gravitational waves propagate. Jun 12 at 2:01
• @user19087 Thanks for pointing it out! I added an extra paragraph trying to clarify this point. If you have any suggestions on how to improve the answer even more, please let me know =) Jun 12 at 2:21
• Re: "those aren't the only sort of waves": To bridge the gap, something about waves created by earth quakes could be included in the answer (with or without addressing the different types). And/or communication/sonar (e.g., whales/dolphins and artificial) to cover liquids not on/near the surface. Jun 12 at 10:41

Spacetime isn't supposed to be a surface, it is solid

It sounds like the idea you were trying for is that space is 3 dimensional, not a 2 dimensional surface. You are right. Spacetime includes time, and is 4 dimensional.

So how do gravitational waves travel through a solid medium?

Let us start with light, because it is a wave that travels through empty space. This will be a classical description of light as opposed to a quantum mechanical description, because gravitational waves are described by general relativity which is a classical theory.

Keep in mind that physics is a mathematical description of the behavior of the universe, not the universe itself. It is never a complete description, because we don't even know the complete behavior. Sometimes we use a simpler classical description, which leaves out some of the behavior. We leave out quantum mechanics here because it is more complicated and usually only makes a difference when describing things the size of atoms and smaller.

If you look carefully at light, it becomes confusing. It is an oscillating electromagnetic field. It can exist in space, and yet space is still empty. So what exactly is it?

I explain a bit in In what medium are non-mechanical waves a disturbance? The aether? The electromagnetic field is a region in space where, if a charge is present, the charge feels an electromagnetic force. If the electromagnetic field oscillates, the charge feels an oscillating force. If no charge is present, there is no force. So in the classical description, nothing happens.

Yet something is going on. If you turn on a light, it takes a while to reach a distant point. A distant charge feels no oscillating force until the light arrives.

At first glance, it is more intuitive in the quantum mechanical description. Photons are waves, or perhaps particles, that travel from the light source outward. But quantum mechanics is notoriously counter intuitive. If you ask what is waving and where is it, the answers are even less clear than "nothing is waving."

Gravitational waves are like this.

And also can they interfere with each other? Please mention any relevant equations

Continuing with light, the relevant classical equations are Maxwell's equations. All by itself, this isn't very helpful. You have to combine two of them to see that they imply a wave equation. If you start with some initial fields distributed in space at some start time, the wave equation predicts the fields at future times. The prediction is that oscillating fields will be present at successively more distant regions of space. That is, electromagnetic waves will travel through space.

Maxwell's equations predict a great many properties of light, including how fast they travel, how big the fields and forces are, and how to generate waves from electric charges.

They predict that two waves interfere with each other. This means that the waves essentially ignore each other and pass through each other unchanged. When they arrive at a charge, the force is the vector sum of the forces from each individual wave. The electric and magnetic fields have a vector nature. They can cancel in one place, and reinforce in a nearby place. The forces also cancel and reinforce.

Gravitational waves are like this too.

Moving a step toward gravitational waves, we need to consider motion at high speed. To understand it, we need a more complete description of the universe. We need to move from classical physics to special relativity.

In classical physics, time moves forward at its implacable rate, and there is nothing you can do about it. Everyone agrees about how much time elapses between two events, even if they disagree about the distance. For example, a man shooting at a target thinks the bullet travels $$100$$ feet from his gun to the target. The bullet thinks it travels $$0$$ feet. Instead the target traveled.

If the bullet (or the target) travels fast enough, it turns out that the same kinds of observer-based disagreements apply to time intervals. To understand them, it is necessary to put time on an equal footing with space, and create spacetime. Instead of a 3 dimensional position vector, you talk about a 4 dimensional vector that describes when you are at a point as well as where that point it.

In special relativity instead of two 3 dimensional E and B vectors, the electromagnetic field is described by a single 4 dimensional anti-symmetric matrix (or anti-symmetric tensor). A 4 dimensional wave equation still predicts that an oscillating electromagnetic field will propagate through time and space. The advantage of this form is that it give correct answers for observers traveling at high speed.

It also explains electric and magnetic fields as parts of a single more complex thing instead of two related simpler things. It explains why you just see an electric field if you stand near a charge. But if you run by it, you also see a magnetic field.

Likewise, time and space are parts of a more complex thing, spacetime.

To understand gravity, we need to go one step farther to general relativity.

As I explain in Why can't I do this to get infinite energy?, time necessarily slows near a massive object.

Distances are distorted as well. A satellite above a planet measures the distance it travels in a single orbit. Dividing by $$\pi$$ gives the diameter. An astronaut descends to the planet, digs through it, and ascends back to orbit. He measures a longer distance and shorter time than expected from classical physics.

As described in What is General Relativity?, these distortions cause geodesics (the "straight" paths followed by objects when there are no forces) to bend. They are the cause of gravity.

Distance and time intervals are described by the metric tensor, a 4 dimensional symmetric matrix. Near a massive object, the metric tensor varies from place/time to place/time.

Like Maxwell's equations describe how charges relate to electric and magnetic fields, the Einstein field equations describes how mass relates to distortions of spacetime. (Actually it is the stress-energy tensor, not mass. Mass and energy are related, as in $$E = mc^2$$.)

Again it takes some work to get a wave equation out of the field equations. But the wave equation predicts that oscillating distortions of spacetime, or gravitational waves, can propagate through spacetime.

Like electromagnetism, this classical description says nothing is waving. But we experience distortions of distance and time as the wave passes. Gravitational wave detectors such as LIGO can measure these incredibly minute distortions. See The Absurdity of Detecting Gravitational Waves

Like electromagnetism, there ought to be a quantum mechanical description that says gravitons travel outward from a source. Unfortunately, so far all attempts to construct a theory of quantum gravity have failed. They lead to predictions of infinitely large values.

This is unfortunate, because we need such a theory to understand the center of black holes and the very early universe. Classical (non-quantum) general relativity predicts an infinitely dense singularity at the center of a black hole. This ignores quantum effects that we know must take place at very small scales, and so is very likely wrong.

• Re "...general relativity which is a classical theory": That is not so clear. Can you elaborate in your answer about the classification choice? (But without "Edit:", "Update:", or similar - the answer should appear as if it was written today.) Jun 12 at 9:56
• @PeterMortensen - Done. I added a couple paragraphs at the end. I didn't use Edit:, though I see no reason why it an unacceptable style. It is not unusual to describe general relativity as classical because it is not quantum. Though there is another sense where classical means Newtonian physics. Jun 12 at 15:38
• I must say that I do not find this very clear. But at the same time I find it as close to being clear as I can imagine anything could be. Thank you Jun 12 at 16:42

space time isn't supposed to be a surface its solid

What exactly is the nature of this comparison ? a solid is a state of matter and a surface is a topological concept. You need to make your question clearer.

Waves can be thought of as disturbances propagating through a medium. While mass / energy are needed in order to generate a gravitational wave, the disturbance itself does not require matter as a medium to propagate through. In the case of gravitational waves the "medium" is spacetime and it is a disturbance in its curvature that propagates. While it isn't as trivial as water or sound waves, it is a prediction of general relativity and it has been observed to be true.

You have ample information in Wikipedia https://en.wikipedia.org/wiki/Gravitational_wave

If you have a solid background and physics and math and are looking to get invested in the topic, I strongly recommend the introductory text

Gravity : An Introduction to Einstein's General Relativity by Hartle

I'm going to take a stab at this with an engineer's basic understanding, and I'll let the better-trained physicists here correct me.

I think, fundamentally, you might be thinking of gravitational waves as a unique phenomenon, when they're not. All you really need to understand to predict them, beyond Newtonian physics, is the fact that information (in this case, a massive object's gravitational effect) can't move through space faster than c. So, when two black holes orbit each other, for example, we don't instantly receive the gravitational effects of each. If we did, we could consider the pair as a point mass, and we wouldn't detect any variation. Instead, we receive the rising and falling gravitational pull of each as they move closer to and further from us, a strength that is inversely proportional to the square of the distance from each black hole to us. Since that information/force/distortion can't move faster than c, the net gravitational effect of the pair rises and falls as they orbit each other, following a sinusoidal pattern. Thus, we call it a wave, simply as an apt descriptor. These effects will be more pronounced if the detector (us) is in the plane of the orbit of the black holes.

In short, if there's nothing weird about gravity propagating in all directions, there's nothing weird about gravitational waves propagating in all directions. We just tend not to naturally think of gravity propagating at c.