# Why curved spacetime produces gravity (am I right?)

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ibkev
I think I may have finally had my eureka moment in understanding how curved spacetime could cause an object at rest to be attracted to another very massive object due to gravity. Could someone please confirm whether what I've written below is correct please?

The unit vectors associated with a given point in a curved spacetime are not orthogonal to one another as they would be in a flat spacetime. This has the effect that motion along one axis couples into one or more other axis because the x,y,z,t coordinates are no longer independent of one another. In this case, the object at rest is still moving through time and because of the curvature of spacetime, this movement is coupled into the spatial dimensions producing gravity.

stevendaryl

Mentor
The unit vectors associated with a given point in a curved spacetime are not orthogonal to one another as they would be in a flat spacetime.

This depends on what unit vectors you choose. You can choose non-orthogonal unit vectors at a point in flat spacetime, or orthogonal unit vectors at a point in curved spacetime. So no, this idea doesn't explain how curved spacetime causes gravitational attraction.

ibkev
ibkev
Thanks Peter... I've got a lot more reading to do. :)

Mentor
Thanks Peter... I've got a lot more reading to do. :)
Search this forum for videos posted by member @A.T. - they are excellent.

ibkev
I think I may have finally had my eureka moment in understanding how curved spacetime could cause an object at rest to be attracted to another very massive object due to gravity. Could someone please confirm whether what I've written below is correct please?

The unit vectors associated with a given point in a curved spacetime are not orthogonal to one another as they would be in a flat spacetime. This has the effect that motion along one axis couples into one or more other axis because the x,y,z,t coordinates are no longer independent of one another. In this case, the object at rest is still moving through time and because of the curvature of spacetime, this movement is coupled into the spatial dimensions producing gravity.

As Peter indicates, you can ALWAYS find an orthogonal (even orthonormal) set of basisvectors. In the orthonormal case they are called the Vielbein. But generally this can only be done locally, not globally as in the flat case.

The basis vectors are linearly independent per definition, so they are always 'independent of one another'.

If you want to understand how gravity (= spacetime curvature) deflects particle trajectories, you have to invoke the concept of a geodesic. The gravitational field obeys Einsteins equations, and the particles in spacetime obey the geodesic equation. Those are two separate things.

ibkev
Staff Emeritus
Imagine drawing a space-time diagram on a flat sheet of paper. Two parallel lines on the piece of paper that are parallel never intersect, never change distance. This space-time diagram represents a pair of particles in an inertial frame with zero relative velocity.

Now imagine drawing a space-time diagram on a curved surface, such as a sphere. The lines on the sphere are as straight as possible, they are the shortest distance between two points. These are called geodesics, and on a sphere they are great circles. Two great circles will always intersect. If you imagine the intersection occurs at the north and south poles of an idealized spherical earth, then the two geodesics will be nearly parallel near the equator. This represents two particles, which are both following natural paths through a curved space-time, that initially have the same relative velocity, but appear to be attracted to each other by tidal forces.

ibkev
The unit vectors associated with a given point in a curved spacetime are not orthogonal to one another as they would be in a flat spacetime. This has the effect that motion along one axis couples into one or more other axis because the x,y,z,t coordinates are no longer independent of one another. In this case, the object at rest is still moving through time and because of the curvature of spacetime, this movement is coupled into the spatial dimensions producing gravity.

That doesn't make sense to me. If space and time were not orthogonal at some point, then you couldn't have an object stay at rest in space at this point, while time passes. But you can have objects stay at rest in a gravitational field, like you now on your chair. The model has to explain all objects, not just free falling ones. What you are missing is the concept of geodesic vs. accelerated worldlines. A free falling object has a geodesic worldline, which can deviate from pure temporal into spatial, even if the dimensions are orthogonal at any point.

ibkev
ibkev
These have been really helpful replies. So my original concept that traveling through time can lead to travel through space when spacetime is curved was in the ballpark but the reason was waay off. :D

When I look at that video the thing that throws me off is what happens to the spatial dimensions of the spacetime grid when it is curved? For example, the tree itself isn't seen to curve because the space it inhabits has curved but I'm not sure why. Do positions within the curved squares shown on the video proportionally correspond to positions with the Euclidean squares?

Straight lines on a flat plane are geodesics. When the plane becomes a curved surface, geodesics stay geodesics (the shortest/longest paths between two points).

That's the conceptual difference between Newton and Einstein. Newton told us that gravity curves particle trajectories in a flat space(time). Instead, Einstein told us that gravity curves spacetime and trajectories are geodesics in this curved spacetime. The fact that geodesics seem to be curved is because the underlying surface is curved.

ibkev
Do positions within the curved squares shown on the video proportionally correspond to positions with the Euclidean squares?
The video strongly exaggerates the amount of space-time distortion you have on Earth's surface, by compressing the time dimension. When drawn in scale (1second ~ 1lightsecond) the time interval would be millions of time longer than the spatial displacement, so the squares would be barely distorted. The spatial curvature around Earth's is also negligible.

ibkev
ibkev
Ok so the effect is so small that in practice it wouldn't be something you could easily measure. This reminded me of the LIGO success so I spent some time watching a bunch of videos for layman describing how it worked.

The brief video below was the best because it was the only one that explicitly mentioned that LIGO couldn't measure the gravitational wave distortion using a ruler made from marks on the rubber sheet of spacetime (they were using for their analogy) because the ruler itself would be warped in the same way that the space it was trying to measure was. So LIGO had to use light.

So if I'm understanding right, the warping of spacetime:
(1) would cause extremely small changes to the spacing between objects within it
(2) would also warp the shape of those objects within it (again extremely small) ie. The objects shape would distort in the same way spacetime distorts, hence the ruler problem LIGO
(3) would introduce new stresses on the objects inside such as tidal forces because the distortions vary with distance.

Is this at least in the ballpark?

So if I'm understanding right, the warping of spacetime:
(1) would cause extremely small changes to the spacing between objects within it
(2) would also warp the shape of those objects within it (again extremely small) ie. The objects shape would distort in the same way spacetime distorts, hence the ruler problem LIGO
(3) would introduce new stresses on the objects inside such as tidal forces because the distortions vary with distance.

Is this at least in the ballpark?

It's seems you describe the effects of changing space-time geometry (like gravitational waves).

ibkev
It's seems you describe the effects of changing space-time geometry (like gravitational waves).

Yes, it's interesting to me that the spacetime we inhabit is curved enough to result in 9.81 m/s^2 gravity yet there is no other effect that we can locally measure. Presumably everything on the Earth is not only subject to this acceleration but is also slightly distorted. From what you're saying the spacetime distortion is very very small yet the resulting gravity is strong enough to hold the moon in orbit.

I think it'll take a while for my intuition here to catchup to the facts! Lol

Yes, it's interesting to me that the spacetime we inhabit is curved enough to result in 9.81 m/s^2 gravity yet there is no other effect that we can locally measure.
As already mentioned, in space-time 1 second along the time dimension corresponds to 1 light-second along the spatial dimensions. The falling apple has to advance a huge distance through time, in order for the small space-time distortion, to create a deviation of a few meters through space.

jimiv
According to my understanding, movement of a body through space-time cause "ripples" in the space-time fabric. If that is so, isn't there some amount of "drag" or "force" of the fabric itself which would tend to slow down that body's inertia?

Mentor
movement of a body through space-time cause "ripples" in the space-time fabric.

Do you have a reference for this? I'm not aware of GR making any prediction of this sort.

jimiv
Do you have a reference for this? I'm not aware of GR making any prediction of this sort.
It was in a documentary I was watching last night (I think a NOVA episode) which was diagramming how gravity is created by large bodies moving through the spacetime continuum. It seemed to suggest that the spacetime fabric itself was being curved by the planetary body moving through it and I was thinking that it must have some type of affect on that body's inertia.

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