# How spacetime distortion can ever be noticed from inside spacetime itself?

this is a naive question from a non-physicist. It is my understanding that gravitational waves are a deformation of spacetime. However, those are noticeable through, for example, laser interferometry. Then I cannot understand how these waves are anything but a variant of electromagnetic waves, i.e. completely generated and detected inside spacetime, without any alteration of the container itself. How can spacetime changes can be noticed from objects that are aware of nothing but their relative interactions with other objects? In other words, if I stretch a napkin on which I have drawn a grid, how does that make a change to the coordinates inside the grid? In this example, it obviously does not, as the coordinates depend on the grid, not something containing it. If I am not clear enough, I will reformulate.

• Probably the question can be broken down into more concrete pieces, seems a bit broad at the moment. – ohneVal Apr 24 at 8:49
• Like posting multiple questions instead? Should I delete this one first if I do? – Exocytosis Apr 25 at 0:12
• Even though it's not specifically about gravitation and spacetime curvature, I think you'll find this recent answer helpful, especially the info about extrinsic curvature. – PM 2Ring Apr 25 at 2:38
• @PM 2Ring: I read but this is not answering my question because I have no problem figuring how to determine from local features that I am on a curved surface or not. Rather my problem is about using the term "spacetime" for something that is not universal. Cells, in biology, perceive their environment through interactions that allow them to move along paths that are not euclidean straight lines but are rather formed by, for example, local differential density of chemical compounds. .../... – Exocytosis Apr 25 at 6:25
• In a way, this fits the idea of geodesics. But even though cells only know those signals and are not aware of three euclidean dimensions, nobody pretends there is no more general space coordinates than theirs. What I mean is, when people say that a gravitational wave is changing spacetime on its way, are they really talking about spacetime or something more relative to its contents? – Exocytosis Apr 25 at 6:25

Quick explanation for the part on gravitational waves would be the following. When one measures something, one always needs a reference, something to compare to, a ruler so to speak. Simplifying things a bit, to measure gravitational waves one uses two laser beams. Vaguely speaking, one compares one with the other. We expect gravitational waves (due to their polarizations) to stretch space in a certain direction but not both (except for subtleties with blind spots). In this way one laser beam will travel longer/shorter in the presence of a gravitational wave while the other remains unaffected, one compares them and concludes space has been perturbed. You can find more details by reading about interferometers.

About the statement saying that they are variants of EM waves, I can say well both phenomena are waves so they are both described by a wave equation, however their nature is very different. EM waves are the things we use currently to measure gravitational waves (i.e laser beams) by means of interferometers, but their polarizations, radiation (methods), mediator particle, spin, etc. are all different. So one has to be very careful when defining an EM wave, this sentence

"completely generated and detected inside spacetime, without any alteration of the container itself."

would imply almost everything is an EM wave.

Now, to address the other part about measuring geometry of space-time while inhabiting space-time, I will point you towards topics but will otherwise be very brief. What follows doesn't pertain directly to gravitational waves but to general relativity.

The point is understanding that certain geometrical properties are indeed recognizable at a local level. That is a surface (more generally a manifold for space-time), can be described locally by means of its metric which is the tool that lets you measure distances and angles at every point (this without the need to imagine your surface as being embedded into a bigger space). The metric then allows the curvature to be computed. The curvature tells us whether our surface locally looks like a sphere or like a saddle or if it looks flat and so on. So I guess your question somehow reduces to how do we measure the curvature of our space-time manifold as inhabitants of said manifold. There is no one way of doing this, but just to mention some ways I will say, by parallel transporting vectors, that is tracking how a certain fixed direction changes while you follow a closed path in your manifold. Another option involves bundles of trajectories called geodesics, where the curvature is related to how they spread from one another while travelling. These are just geometrical facts upon which general relativity is built on.

Your example with the napkin is very specific and enjoys a lot of symmetry, so it doesn't really serve to counter your intuition on measurements. If you expand a napkin with a grid (with which you are probably thinking about the universe expanding) you are implicitly saying that your objects are fixed to the grid which is not how we would know the napkin is expanding. It is true that spatial features like distances and angles will remain the same (see conformal transformations), that is things that are fixed to the grid (comoving objects) won't notice any change in relative distances or angles. But about things that are not fixed to the grid (like light beams) we can say a lot, think about using the Doppler shift for measuring receding/approaching speeds.

If you are really interested you might want to learn and read more on differential geometry and general relativity at the formal level. For which you can start in Wikipedia and then move on to the references therein. I hope this brief explanation serves its purpose.

• First thank you for taking the time to write this long answer. I don't understand why you write that light is not fixed to the coordinate system (the grid). How could that be? I don't suppose you refer simply to the fact light particles are moving, as motion is allowed on a cartesian space, but to something else. You mentioning metrics make me wonder if my question can be reduced to what spacetime are we talking about. .../... – Exocytosis Apr 25 at 0:08
• Is spacetime locally defined by local properties with global dimensions of space and time being only macrophenomena that can only be measured by integrating infinitesimal local variations of spacetime structure? I think what is misleading me is actually wikipedia because its gravitational wave page shows animations of rings of dots perturbated by said waves and wobbling (page section "effects of passing"). So the picture looks like the coordinate system, ie. grid, is static (horizontal and vertical dimensions of the video do not vary), while only the dots inside the picture move. .../... – Exocytosis Apr 25 at 0:09
• If the positions of the dots are the spacetime, one may wonder what the horizontal and vertical dimensions of the picture are. – Exocytosis Apr 25 at 0:09
• What I meant is that light doesn't have a restframe, a photon is never at rest. So light is very special in that regard, no matter what coordinates you use it will always be perceived to be moving at "light speed". About the second and third comment, it is indeed space, the dots can be taken to move in the $x-y$-plane when a gravitational wave propagates in $z$. So the positions are not the spacetime, the positions ARE the actual spatial positions, there is nothing more than that in those animations. – ohneVal Apr 25 at 8:37
• But if a wave propagating in the Z direction is able to displace particles in the perpendicular plane, it does not require, per se, to alter spacetime, does it? I mean, if you send a pressure wave through a solid, you might get the same result. Now I know a phonon would not propagate through vacuum, but what aspect of gravitational waves prevent from stating that they are simply some sort of phonons able to travel through space? .../... – Exocytosis Apr 26 at 12:50