# Spacetime changes vs location of matter inside

• I
Putting aside gravity, if there were some way to stretch and twist spacetime, what effect would that have on the matter inside the volume? Say we were talking about a part of spacetime that enclosed the earth, would the earth itself be stretched and twisted? Also, wouldn't a ruler I used be warped in the same way as any object that it might be used to measure and therefore there would be no detectable effect?

Also, if I where able to expand spacetime (which is what I understand inflation to be) would the matter inside it similarly expand? E.g. Say there were 2 objects inside the expanding spacetime, would they get farther apart and also become larger themselves?

PeterDonis
Mentor
Putting aside gravity, if there were some way to stretch and twist spacetime

This doesn't seem like a consistent hypothesis, so I'm not sure what you're thinking about makes sense. However, see below.

wouldn't a ruler I used be warped in the same way as any object that it might be used to measure

No. Spacetime curvature doesn't "warp rulers" (although some pop science treatments misleadingly describe it in terms that might make you think it does). Spacetime curvature is another name for tidal gravity. You can measure tidal gravity solely using freely falling objects that are not being subjected to any forces, so they aren't "warped" by anything.

if I where able to expand spacetime (which is what I understand inflation to be) would the matter inside it similarly expand?

No.

• Dale and ibkev
Thanks Peter!

If I have a box with a bunch of stuff in it (as an analogy to spacetime with some distribution of mass in it) and then I stretch/squish the box into a different shape, doesn't the distribution of mass inside the box necessarily have to change to accommodate this new, enclosing shape?

(Does that question even make sense? :)

PeterDonis
Mentor
If I have a box with a bunch of stuff in it (as an analogy to spacetime with some distribution of mass in it) and then I stretch/squish the box into a different shape, doesn't the distribution of mass inside the box necessarily have to change to accommodate this new, enclosing shape?

Yes. But this is not the same thing as stretching or squishing spacetime, although it can be related to the latter, since changing the distribution of mass can also change the spacetime geometry, because the two are related via the Einstein Field Equation. But they're not the same.

• ibkev
I found a question on the physics.stackexchange that phrased the question I wanted to ask much better than I did.

Suppose you have a metal cube floating in space far from massive objects so spacetime is flat and you measure the sides of the cube with a ruler to be 1m. Now you transport the cube to the surface of a planet where spacetime is strongly curved. Has the size/geometry of the cube objectively changed in a way you could measure with say a laser and if so, does the change in the cube's geometry match the curvature of the spacetime?

Does the ruler still measure the cube as 1m for all sides because both the cube and the ruler are warped by the curved space?

PeterDonis
Mentor
Has the size/geometry of the cube objectively changed in a way you could measure with say a laser

It depends.

If you let the cube be in free fall, and the cube is small enough that tidal gravity is negligible over its size, then no.

If the cube's center of mass is in free fall, but tidal gravity is large enough over its size to produce measurable stresses in the cube, then its size/geometry might change, depending on its tensile strength vs. the size of the stresses.

If the cube is accelerated, i.e., it is sitting at rest on the surface of the planet, not in free fall, then in general its geometry will change; but the change will be different from the tidal gravity case above, because it is due to the cube's proper acceleration, i.e., to the weight of the cube's upper parts pressing down on its lower parts.

does the change in the cube's geometry match the curvature of the spacetime?

Not really. The way you measure the geometry of spacetime is by measuring tidal gravity; but the changes in the cube's geometry are, at best, only indirectly related to tidal gravity (in the second case above), and possibly not related to it at all (in the third case above).

Does the ruler still measure the cube as 1m for all sides because both the cube and the ruler are warped by the curved space?

This depends on what you make the ruler out of. The ruler is a physical object just like the cube, so it can be distorted just like the cube is, based on the factors above. But there are ways to get around that. For example, to measure the size of a cube that is accelerated, have the ruler in free fall, and take the length measurement when the ruler is momentarily at rest relative to the cube. Or, you can use lasers, as you said.

One thing to note, though, is that in general, different methods of measuring lengths, even if all precautions are taken, will not necessarily give the same results in a curved spacetime, or even in flat spacetime if the measurements are being made on accelerated objects. So the spatial "size" of an object is not quite an invariant property of the object the way our intuitions say it ought to be.

• ibkev
Many thanks for this reply Peter!