Question about gravity being a distortion of spacetime

[lordoftheselands]

TL;DR Summary

gravity

people say that gravity is not a force, that it's rather a distortion of space-time

so objets that go to a gravitational center are actually just moving through space in linear direction

but there is a problem in this theory

shouldn't objects go to the center in constant speed? why are they being accelerated?

 

[Dale]

why are they being accelerated?

They are not being accelerated in an inertial frame. If you attach an accelerometer you will see that it reads 0 at all times.

Of course, you can use non-inertial frames if you want and that will introduce a fictitious force with coordinate acceleration in that frame, but as always fictitious forces are not detected with accelerometers.

 

[Ibix]

so objets that go to a gravitational center are actually just moving through space in linear direction

They are following a geodesic, which is the generalisation of a straight line in curved geometry, in spacetime. When you project that on to "space" (a surface of constant Schwarzschild coordinate time, usually) the rate of travel varies because both space (by that definition) and spacetime are curved.

 

[pervect]

TL;DR Summary: gravity

people say that gravity is not a force, that it's rather a distortion of space-time

so objets that go to a gravitational center are actually just moving through space in linear direction

but there is a problem in this theory

shouldn't objects go to the center in constant speed? why are they being accelerated?

Consider two great circles on the curved surface of the Earth.

The great circles are like straight lines, in that they are the shortest distance between two points.

If you have two ships that follow great circles routes from the north pole to the south pole, they will initially move away from each other as judged by the rate of change of their separation vector. The rate of change of their separation decreases, eventually stopping at the equator, and then they start to approach each other.

This is an example of what is called "geodesic deviation", great circles on the surface of a sphere, and straight lines in the Euclidean plane, are examples of geodesics.

The mathematics of curvature say that geodesics on curved surfaces accelerate away from each other. In fact, that's one of the possible definitions of curvature.

The example of the Earth's surface is an example of a curved spatial surface, which is easier to visualize. One can gain some insight into GR by imagining that one draws space-time diagrams on a curved surface, such as a sphere, though this technique can only handle 2 dimensons, one of space and one of time, and not the 4 dimensions neaded for actual space-time. To really compute results, one needs to go beyond such simple visualizations and treat the topic mathematically.

In our simple great circle example, then, it's better to imagine one of the dimensions as time, say the north-south motion, and the other dimension (east-west) as space.

The great circle example lacks the feature of how matter determines geometry - the geometry of the sphere is just given. In GR, there is a mathematical relationship between the distribution of matter (energy, momentum, and pressure), and the curvature of space-time given by Einstein's field equations.

Sadly, both the term on the left side (related to curvature) and the term on the right hand side (related to matter distribution) are not easy to discuss without a considerable amount of background. The thing on the left hand side is called the "Einstein curvature tensor", and the thing on the right hand side is called "the stress energy tensor", but the names won't mean much without physics and maths that is usually introduced at the graduate level (or sometimes the late undergraduate level).

 

[A.T.]

shouldn't objects go to the center in constant speed?

Why do you think so? See the animation below for why their speed in the non-falling frame increases, just like in Newtonian Gravity. In distorted space-time, the free-fall world-line deviates more and more towards the spatial (downwards) dimension.

 

[PeterDonis]

objets that go to a gravitational center are actually just moving through space in linear direction

No, they are moving through spacetime along geodesics. Geodesics are "straight lines" in curved spacetime, but that in no way implies that they are straight lines in space.

 

[Ibix]

that in no way implies that they are straight lines in space.

And, to add to this, even when they are straight lines in space (as in the case of radial infall) it does not necessarily imply that they have a uniform velocity in space.

 

[Ibix]

Here's a plot of the worldline of an infalling object in Schwarzschild coordinates:

Time goes up the page and the $r$ coordinate increases to the right. The event horizon would be a vertical line at $r/R_S=1$ in this representation (and the status of the graph to the left of that is a bit dubious, although that doesn't matter here). The object was released from rest at twenty times the Schwarzschild radius. It falls into the hole along the blue path with a clock whose ticks (at intervals of $5R_S/c$) are marked as crosses. The sharp upward turn at the top left is where the gravitational time dilation begins to dominate and the downward progress (as measured in these coordinates) slows.

The blue line is a geodesic - it is straight. The reason it is shown as curved is because the lines of constant $r$ and lines of constant $t$ that would make a rectangular grid on this diagram are actually curved (they are not geodesics) and I've picked a coordinate system that renders them straight but bends the geodesic. But this coordinate system is pretty much the naive understanding of "space" and "time" and the weak field in these coordinates reduces directly to Newtonian gravity. That's why I picked these coordinates, and the curviness of the gridlines is why we see acceleration when we look at it naively.

I could pick a different coordinate system that straightens out the blue line (I think Gullstrand-Painleve coordinates do it, although that may only be for objects falling from infinity - I'd need to check). However, the price to be paid is that the ingoing and outgoing speeds of light are not symmetric and hovering clocks have a more complicated relationship to clocks at infinity.