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Layman's question: are gravitational waves really "waves"? When I think of waves, I think of systems governed by the wave equation [0]. But IIUC that requires

Layman's question: are gravitational waves really "waves"? When I think of waves, I think of systems governed by the wave equation [0]. But IIUC that requires a restoring force. I'm not sure what that would mean in the case of gravity. E.g., if the moon suddenly lurched towards the Earth, we'd perceive an increase in gravity between the two. But that would be a semi-permanent change in the strength of that attraction between the two objects; not what I'd think of as a wave-like fluctuation. [0] https://en.wikipedia.org/wiki/Wave_equation https://en.wikipedia.org/wiki/Wave_equation

joe__f 5y ago
Yes, gravitational wave are governed by the wave equation. A standard way to treat them is as a small perturbation around a known spacetime geometry. Then the metric would be $g = g_0 + \epsilon h$, where $g_0$ is the background, $\epsilon$ is a small parameter and $h$ is the perturbation. In general Einstein's field equations govern the dynamics of $g$, and if you take the first order behaviour in $\epsilon$ around $g_0$ as flat space, then you recover the wave equation for $h$ as in the article you linked, with propagation speed $c$ the speed of light. (There are some additional subtleties about choice of gauge, but this is not physical). I'm not sure what you mean with your moon example.
- CoastalCoder 5y ago
  Thanks! I'm not ready to understand your explanation, but I can clarify the moon example at least. Suppose the gravitational attraction between the earth and moon is 2e20 Newtons. Now imagine something forces that attraction to strengthen from 2e20N to 3e20N. E.g. the Earth and moon get closer to each other, or the moon gets more massive somehow. When we talk about "detecting gravity waves", I understand that to mean that on Earth we've managed to detect that increase from 2e20N to 3e20N. My point was: perhaps that looks like the rising edge of a wave phenomenon, but it's not actually cyclic like I expect from a traditional "wave". So I couldn't understand why it's called a gravity wave.
  - Beldin 5y ago
    Nitpick: gravity waves != gravitational waves. Unfortunately, I don't know enough to explain the difference.
    - cobaltoxide 5y ago
      "Gravity waves" are ordinary water waves, or other waves of a fluid, where gravity provides the restoring force: https://en.wikipedia.org/wiki/Gravity_wave https://en.wikipedia.org/wiki/Gravity_wave They have absolutely nothing to do with gravitational waves. Nonetheless, in the gravitational wave community (i.e. LIGO) we don't usually ever talk about true "gravity waves" so we use the terms interchangably. The exception is that "gravity waves" in the Earth and the atmosphere can couple to LIGO through ordinary Newtonian gravity, which is something we have to think about and exclude.
  - cobaltoxide 5y ago
    Gravitational waves do not behave in the way you have described. > I understand that to mean that on Earth we've managed to detect that increase from 2e20N to 3e20N. This not what how gravitational waves behave. This assumption is not correct and will confuse you.
  - xelxebar 5y ago
    If you're already comfortable with the Wave Equation, then it's somewhat straightforward. You start with some spacetime geometry (i.e. some "known metric") and then study small perturbations that obey the wave equation. The key is finding distributions of matter that, by Einstein's Equation, cause such perturbations to arise. In differential geometry, the device called a "metric" let's you define things like angles and distances on some space. This is generally denoted as g with the Greek letters mu and nu as subcripts. Just to give you a feel of what's going on, I'll leave out the subscripts, but what OP's saying is that g = k + h which says that g, the space-time metric, decomposes into some known, usually easy to work with metric k, plus some perturbations on top, h. In the math you put restrictions on h to make precise the "small perturbations" part. Anyway, what Gravity Waves boil down to are situations (read distributions of matter) that cause h to obey the wave equation. In other words, gravitational waves are spacetimes that have a component that waves "on top" of a given reference "background" spacetime. Of course, you could cherry pick your background such that anything looks like waves on top, but typically we choose the background to be Minkowskian (i.e. flat). FWIW, the above perturbative approach is called Linearized Gravity and is kind of a (nice) hack to restrict study of the really hairy non-linear Einstein Equation to cases that are manageable.
cobaltoxide 5y ago
Gravitational waves are waves of the metric tensor. Einstein's field equations provide the "restoring force." Importantly, gravitational waves are not waves of Newtonian gravity. Gravitational waves do not "push and pull" along the direction of propagation. They stretch and compress space along axes perpendicular to the direction of propagation. > E.g., if the moon suddenly lurched towards the Earth, we'd perceive an increase in gravity between the two. But that would be a semi-permanent change in the strength of that attraction between the two objects; not what I'd think of as a wave-like fluctuation. Indeed, gravitational waves do not work this way. Unfortunately it is hard to explain gravitational waves without significant math. In fairness, even Einstein himself waffled over whether gravitational waves would be a real effect predicted by the theory. Then it took nearly a century to detect them experimentally, and there were plenty of doubters along the way. The results one gets from intuition are generally incorrect in important ways. Here's a derivation of a wave equation from Einstein's field equations: https://en.wikipedia.org/wiki/Linearized_gravity https://en.wikipedia.org/wiki/Linearized_gravity A more helpful introduction for a layperson might be the paper titled "Gravitational Waves on the back of an envelope": https://aapt.scitation.org/doi/10.1119/1.13627 https://aapt.scitation.org/doi/10.1119/1.13627 But, infuriatingly, that paper does not seem to be open-access. Here's someone's scanned copy: https://www.ru.ac.za/media/rhodesuniversity/content/mathematics/images/Schutz1984-AJP000412-1.pdf https://www.ru.ac.za/media/rhodesuniversity/content/mathemat...
- fsflover 5y ago
  > But, infuriatingly, that paper does not seem to be open-access. Try sci-hub.
- sildur 5y ago
  > They stretch and compress space along axes perpendicular to the direction of propagation. Is it possible to compress space?
  - drran 5y ago
    (Not a native speaker). Space is a mathematical abstraction. You can do anything with, because it's just math. If you want to bend, curve, distort, rip space, or add more dimensions — go for it. Physical medium should be compressible, like any other medium. Higgs «field» (let's call it Higgium — Higgs+Vacuum) is presented everywhere, because Higgs boson gives mass to every particle in the Visible Universe and beyond, so it should conduct gravitational waves. As demonstrated by LIGO/VIRGO, gravitational waves causes distortions in light travel. These distortions can be explained as changes of conductivity in Higgium, which can be caused by changes in density, so yes, it can be waves of compression.
akomtu 5y ago
I'm hardly an expert in GR, but it seems to me that gravitational waves is a trivial consequence of limited speed of gravity propagation. It takes 8 mins for Sun's gravity to reach Earth. If Sun suddenly moves left and right, in 8 mins we'll feel a pull to the left and then to the right, so in effect gravity will feel like a couple waves passing thru us. The exact geometry of those waves is complex, but that hardly matters for intuitive understanding. The exact geometry of water ripples is also very complex, after all.
- jleahy 5y ago
  Unfortunately that’s not how they behave at all. See some of the references posted by others in other comments. Intuition doesn’t really work at all here.
  - cobaltoxide 5y ago
    Agree with jleahy. I would add that considering counterfactual situations like "if the sun were to move instantly to the right" is of limited utility. That cannot happen (violates conservation of momentum). (Granted, other thought experiments have been quite influential in this field...)
- contravariant 5y ago
  Apart from some tidal forces we probably wouldn't feel a thing but rather just continue following the free falling trajectory. Gravitational waves are also different from changes to the amount of gravitational force. This can be seen from the fact that gravitational waves affect the distances between objects, whereas the sun which exerts far more gravitational force has no measurable effect (at least, to my knowledge rotating objects don't change length noticeably).
- c1ccccc1 5y ago
  I think it's actually more complicated than that. The strength of the sun's gravity goes as 1/r^2. Move it to the side a bit, and the direction of the gravitational field changes slightly. This direction change goes as 1/r, by the small angle approximation. So from your description, we'd expect the size of the ripple to go as 1/r^3. But in the case of actual gravitational waves, the field strength of the ripple goes as 1/r. That's why we can detect gravitational waves from distant black holes, but we can't detect their static gravitational fields, which go as 1/r^2. Electromagnetic waves are similar. The Lienard-Wiechert field formulas [1] have 2 terms. The first term describes the delayed field, and is proportional to 1/r^2, while the second describes waves and is proportional to 1/r. [1] https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential#Field_computation https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_...

joe__f 5y ago

Yes, gravitational wave are governed by the wave equation. A standard way to treat them is as a small perturbation around a known spacetime geometry. Then the metric would be $g = g_0 + \epsilon h$, where $g_0$ is the background, $\epsilon$ is a small parameter and $h$ is the perturbation. In general Einstein's field equations govern the dynamics of $g$, and if you take the first order behaviour in $\epsilon$ around $g_0$ as flat space, then you recover the wave equation for $h$ as in the article you linked, with propagation speed $c$ the speed of light. (There are some additional subtleties about choice of gauge, but this is not physical). I'm not sure what you mean with your moon example.

CoastalCoder 5y ago
Thanks! I'm not ready to understand your explanation, but I can clarify the moon example at least. Suppose the gravitational attraction between the earth and moon is 2e20 Newtons. Now imagine something forces that attraction to strengthen from 2e20N to 3e20N. E.g. the Earth and moon get closer to each other, or the moon gets more massive somehow. When we talk about "detecting gravity waves", I understand that to mean that on Earth we've managed to detect that increase from 2e20N to 3e20N. My point was: perhaps that looks like the rising edge of a wave phenomenon, but it's not actually cyclic like I expect from a traditional "wave". So I couldn't understand why it's called a gravity wave.
- Beldin 5y ago
  Nitpick: gravity waves != gravitational waves. Unfortunately, I don't know enough to explain the difference.
  - cobaltoxide 5y ago
    "Gravity waves" are ordinary water waves, or other waves of a fluid, where gravity provides the restoring force: https://en.wikipedia.org/wiki/Gravity_wave https://en.wikipedia.org/wiki/Gravity_wave They have absolutely nothing to do with gravitational waves. Nonetheless, in the gravitational wave community (i.e. LIGO) we don't usually ever talk about true "gravity waves" so we use the terms interchangably. The exception is that "gravity waves" in the Earth and the atmosphere can couple to LIGO through ordinary Newtonian gravity, which is something we have to think about and exclude.
- cobaltoxide 5y ago
  Gravitational waves do not behave in the way you have described. > I understand that to mean that on Earth we've managed to detect that increase from 2e20N to 3e20N. This not what how gravitational waves behave. This assumption is not correct and will confuse you.
- xelxebar 5y ago
  If you're already comfortable with the Wave Equation, then it's somewhat straightforward. You start with some spacetime geometry (i.e. some "known metric") and then study small perturbations that obey the wave equation. The key is finding distributions of matter that, by Einstein's Equation, cause such perturbations to arise. In differential geometry, the device called a "metric" let's you define things like angles and distances on some space. This is generally denoted as g with the Greek letters mu and nu as subcripts. Just to give you a feel of what's going on, I'll leave out the subscripts, but what OP's saying is that g = k + h which says that g, the space-time metric, decomposes into some known, usually easy to work with metric k, plus some perturbations on top, h. In the math you put restrictions on h to make precise the "small perturbations" part. Anyway, what Gravity Waves boil down to are situations (read distributions of matter) that cause h to obey the wave equation. In other words, gravitational waves are spacetimes that have a component that waves "on top" of a given reference "background" spacetime. Of course, you could cherry pick your background such that anything looks like waves on top, but typically we choose the background to be Minkowskian (i.e. flat). FWIW, the above perturbative approach is called Linearized Gravity and is kind of a (nice) hack to restrict study of the really hairy non-linear Einstein Equation to cases that are manageable.

cobaltoxide 5y ago

Gravitational waves are waves of the metric tensor. Einstein's field equations provide the "restoring force." Importantly, gravitational waves are not waves of Newtonian gravity. Gravitational waves do not "push and pull" along the direction of propagation. They stretch and compress space along axes perpendicular to the direction of propagation. > E.g., if the moon suddenly lurched towards the Earth, we'd perceive an increase in gravity between the two. But that would be a semi-permanent change in the strength of that attraction between the two objects; not what I'd think of as a wave-like fluctuation. Indeed, gravitational waves do not work this way. Unfortunately it is hard to explain gravitational waves without significant math. In fairness, even Einstein himself waffled over whether gravitational waves would be a real effect predicted by the theory. Then it took nearly a century to detect them experimentally, and there were plenty of doubters along the way. The results one gets from intuition are generally incorrect in important ways. Here's a derivation of a wave equation from Einstein's field equations: https://en.wikipedia.org/wiki/Linearized_gravity https://en.wikipedia.org/wiki/Linearized_gravity A more helpful introduction for a layperson might be the paper titled "Gravitational Waves on the back of an envelope": https://aapt.scitation.org/doi/10.1119/1.13627 https://aapt.scitation.org/doi/10.1119/1.13627 But, infuriatingly, that paper does not seem to be open-access. Here's someone's scanned copy: https://www.ru.ac.za/media/rhodesuniversity/content/mathematics/images/Schutz1984-AJP000412-1.pdf https://www.ru.ac.za/media/rhodesuniversity/content/mathemat...

fsflover 5y ago
> But, infuriatingly, that paper does not seem to be open-access. Try sci-hub.
sildur 5y ago
> They stretch and compress space along axes perpendicular to the direction of propagation. Is it possible to compress space?
- drran 5y ago
  (Not a native speaker). Space is a mathematical abstraction. You can do anything with, because it's just math. If you want to bend, curve, distort, rip space, or add more dimensions — go for it. Physical medium should be compressible, like any other medium. Higgs «field» (let's call it Higgium — Higgs+Vacuum) is presented everywhere, because Higgs boson gives mass to every particle in the Visible Universe and beyond, so it should conduct gravitational waves. As demonstrated by LIGO/VIRGO, gravitational waves causes distortions in light travel. These distortions can be explained as changes of conductivity in Higgium, which can be caused by changes in density, so yes, it can be waves of compression.

akomtu 5y ago

I'm hardly an expert in GR, but it seems to me that gravitational waves is a trivial consequence of limited speed of gravity propagation. It takes 8 mins for Sun's gravity to reach Earth. If Sun suddenly moves left and right, in 8 mins we'll feel a pull to the left and then to the right, so in effect gravity will feel like a couple waves passing thru us. The exact geometry of those waves is complex, but that hardly matters for intuitive understanding. The exact geometry of water ripples is also very complex, after all.

jleahy 5y ago
Unfortunately that’s not how they behave at all. See some of the references posted by others in other comments. Intuition doesn’t really work at all here.
- cobaltoxide 5y ago
  Agree with jleahy. I would add that considering counterfactual situations like "if the sun were to move instantly to the right" is of limited utility. That cannot happen (violates conservation of momentum). (Granted, other thought experiments have been quite influential in this field...)
contravariant 5y ago
Apart from some tidal forces we probably wouldn't feel a thing but rather just continue following the free falling trajectory. Gravitational waves are also different from changes to the amount of gravitational force. This can be seen from the fact that gravitational waves affect the distances between objects, whereas the sun which exerts far more gravitational force has no measurable effect (at least, to my knowledge rotating objects don't change length noticeably).
c1ccccc1 5y ago
I think it's actually more complicated than that. The strength of the sun's gravity goes as 1/r^2. Move it to the side a bit, and the direction of the gravitational field changes slightly. This direction change goes as 1/r, by the small angle approximation. So from your description, we'd expect the size of the ripple to go as 1/r^3. But in the case of actual gravitational waves, the field strength of the ripple goes as 1/r. That's why we can detect gravitational waves from distant black holes, but we can't detect their static gravitational fields, which go as 1/r^2. Electromagnetic waves are similar. The Lienard-Wiechert field formulas [1] have 2 terms. The first term describes the delayed field, and is proportional to 1/r^2, while the second describes waves and is proportional to 1/r. [1] https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential#Field_computation https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_...