# Acceleration from curved spacetime

## Main Question or Discussion Point

This might be a stupid question..

Why does curved spacetime cause objects with mass to accelerate towards each other?

If I placed a massive particle next to a larger massive object, at rest with respect to the large object, shouldn't the particle stay at rest?

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Matterwave
Gold Member
Er...

GRAVITY causes masses to accelerate towards each other. In a Newtonian framework, gravity is a FORCE, which causes objects to accelerate towards each other according to F=ma.

In a General Relativistic framework, gravity is a curvature in space-time. Particles follow trajectories in space-time (also called world-curves), e.g. (ct, x(t), y(t), z(t)) and these world curves are determined by the geodesics of the space-time. You will find that the geodesics or world curves are such that massive objects are accelerated towards each other (almost the same way they are in Newtonian gravity, with larger and larger discrepancies occurring for larger and larger curvatures).

DrGreg
Gold Member
This might be a stupid question..

Why does curved spacetime cause objects with mass to accelerate towards each other?

If I placed a massive particle next to a larger massive object, at rest with respect to the large object, shouldn't the particle stay at rest?
In its simplest form, we can consider just one dimension of space which gives us a two-dimensional spacetime, which is just a distance-against-time graph.

Without gravity, we draw the graph on flat paper. Objects moving freely (with no force on them) have graphs that are straight lines.

With gravity, we draw the graph on a curved surface. Two objects that begin at rest relative to each other have worldlines (graph lines) that are initially parallel, but the curved surface causes the lines to converge or diverge further along.

Thanks Matterwave and DrGreg.

DrGreg, I was thinking about curved spacetime as in the picture on the link you gave when I asked my question. I understand how a moving object follows a straight line along the curved spacetime. What I am wondering about is why an object that is placed at rest (say, 1 meter above the earth) would accelerate towards the earth (I guess both would be accelerating towards each other). For example, in the curved spacetime cone in the picture, the placed object is at rest 1 meter away from the earth, and since it is not moving, it isn't following the curvature toward earth. So why does it accelerate toward earth?

Does curved spacetime cause the object to sort of slide toward earth? Jonathan Scott
Gold Member
You're thinking about curved space, not curved space-time. Curved space only causes moving things to accelerate. Curved space-time is also curved (typically by approximately the same amount) with respect to time, so when you plot position in space against time in equivalent units you get the same curvature, corresponding to an acceleration.

Cleonis
Gold Member
and since it is not moving, it isn't following the curvature toward earth.
Well, in terms of relativistic physic it is moving. Your time coordinate is always increasing.

In relativistic physics the word 'motion' does not mean the same thing as in classical physics.

Cleonis
Gold Member
What I am wondering about is why an object that is placed at rest (say, 1 meter above the earth) would accelerate towards the earth.
Let me recast that question.

The background.
When you throw an object upwards then at the point in time where it reaches its highest point it has momentarily zero spatial velocity (relative to you, the thrower).

In a sense you are asking why objects, when they have reached their highest point, fall back again.

www.relativitet.se/spacetime1.html

Does curved spacetime cause the object to sort of slide toward earth?
No, thinking along lines like that is wrongfooting yourself.
What you seem to think of is the cone as an inclined surface, with a force causing objects placed on it to slide in some direction.
The image of the cone does not have that purpose at all.

Here the entire base of the cone represents a single point. You throw an object straight upwards, and it fall right back in your hand.

The line along the surface of the cone has the following property. At every point along the way the line does not curve right or left relative to the local surface. The line curves back down because of the curvature of the surface as a whole.

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Yeah, I think my problem is that I'm not considering the time component of spacetime..

It's difficult for me to conceptualize curved spacetime. Even the cone is confusing to me because the time coord wraps around to meet back at the original point.

Anyway, I'll keep thinking about it. Thanks for the help!

It IS confusing...

This reminded me that in 2+1 dimensional (2 space and time) constant velocity is a straight line, linear acceleration is a curve and rotational acceleration a corkscrew....so I think that's the physical sense in which acceleration in Minkowski space causes a rotation??? and why acceleration must be perpendicular to velocity...strange as that seems at first....
If so I now have a physical picture for:

...
we can regard acceleration in spacetime as simply a rotation of the 4-velocity
and which reminded me rotation in Euclidean SPACE and Minkowski SPACETIME don't look the same....

And you will note that in spacetime, velocity is constant (c)....and acceleration doesn't change that speed...it changes the shape of the worldline thru spacetime....

DrGreg
Gold Member
Yeah, I think my problem is that I'm not considering the time component of spacetime..

It's difficult for me to conceptualize curved spacetime. Even the cone is confusing to me because the time coord wraps around to meet back at the original point.
The "cone" shape is an analogy, and like all analogies you can't push it too far. Time doesn't really go in a circle, so the analogy isn't perfect.

DrGreg
Gold Member
Isn't it easier to visualize from this diagram:

http://en.wikipedia.org/wiki/Spacetime
The problem with that diagram is that it depicts only the curvature of space, not of spacetime. As such it doesn't really explain gravity -- you need to consider the time dimension too.

Dr Greg..I was afraid somebody would say that....

why is it that we can't think of that diagram in terms of the explanation given right under it:

White lines do not represent the curvature of space but instead represent the coordinate system imposed on the curved spacetime, which would be rectilinear in a flat spacetime.

DrGreg
Gold Member
Dr Greg..I was afraid somebody would say that....

why is it that we can't think of that diagram in terms of the explanation given right under it:
Well, contrary to what the caption says, I think it does illustrate the curvature of space, but that alone doesn't explain gravity. Why would an object released from rest fall to Earth? The diagram would explain that only if there was a gravitational force acting downwards on the entire diagram, but where would such a force come from?

pervect
Staff Emeritus
This might be a stupid question..

Why does curved spacetime cause objects with mass to accelerate towards each other?

If I placed a massive particle next to a larger massive object, at rest with respect to the large object, shouldn't the particle stay at rest?
First of all, you should probably simplify the problem by thinking of it in terms of test particles. Therefore, curved space-time, the curvature being due to mass (and perhaps to other things in GR that can cause curvature), will cause very-low-mass test particles near the massive object to follow a curved path as if they were attracted to said object by some force.

Note the difference - when you start to talk about particles themselves curving space-time, the picture becomes a lot more complicated, because you have to take into account all sorts of back-reactions as the massive particles move. When you think about particles of low enough mass that they have an insignificant effect on curvature, used to probe the structure of a static space-time around some simple massive object (like a black hole), it becomes a lot easier situation to analyze.

The other point you are most likely missing is that it's not space that's curved, it's space-time. Take a snapshot of the particle relative to the large mass at some "instant in time", it has a certain distance and a certain velocity. Take another snapshot of the particle relative to the large mass at some later "instant in time". Now it is both closer to the large mass, and has a velocity that points towards the large mass. The conclusion is that the mass has somehow "attracted" the particle, even though it's simply been following a geodesic.

The notion of what particular surface, i.e. set of points, of space-time, that corresponds to an "instant in time" is observer dependent - the easiest solution is to use a very slowly moving particle (perhaps one that is even initially not moving at all), in which case the notion of time experienced by said test particle is the same as the notion of time defined by the geometry of the large mass.

If you use a short time interval, you'll see that the distance moved by an initally stationary particle is quadratic in time, 1/2 a t^2, so to first order in time it's zero, while the acquired velocity is linear, a t, and that's what you expect the geometry to reproduce.

The sort of diagrams you really need to draw to find the velocity cab be found in
Donald Marolf, "Spacetime Embedding Diagrams for Black Holes", http://arxiv.org/abs/gr-qc/9806123. The diagarams that show spatial curvature are interesting, but won't help you understand the answer to your question.

However, you'll need to be familiar with the space-time diagrams of special realtivity and the Lorentz transform to really understand Marolf's particular embedding fully, and the article itself is written from a rather technical viewpoint.

DrGreg:

Well, contrary to what the caption says, I think it does illustrate the curvature of space, but that alone doesn't explain gravity. Why would an object released from rest fall to Earth? The diagram would explain that only if there was a gravitational force acting downwards on the entire diagram, but where would such a force come from?
I think I understand what you mean. It is "artificial". But I still like the grid in Wiki more than you.

The explanation you provided me in the thread I referenced above gave me another insight into how I am still not "visualizing" spacetime effects as accurately as I would like. It's confusing, I suspect for Jorjy and many of us, because we are used to acceleration changing speed,not time and yet in Einstein's spacetime acceleration acts analogous to rotational motion in space (where aceleration is perpendicular to velocity), speed remains constant while direction changes, but acceleration in spacetime also changes passage through TIME...the faster one moves the slower the passage of time for them....

The best way so far for me to "visualize" that is that linear acceleration in space appears as a curve in spacetime.....

pervect posts

The notion of what particular surface, i.e. set of points, of space-time, that corresponds to an "instant in time" is observer dependent - the easiest solution is to use a very slowly moving particle (perhaps one that is even initially not moving at all), in which case the notion of time experienced by said test particle is the same as the notion of time defined by the geometry of the large mass.
Why would the notion of time of the test particle and that of the mass be the same when the gravitational fields (potential) are different? I assume you imply here the relative velocities of the mass and the test particle are the same....??

DrGreg
Gold Member
I think I understand what you mean. It is "artificial". But I still like the grid in Wiki more than you.

The explanation you provided me in the thread I referenced above gave me another insight into how I am still not "visualizing" spacetime effects as accurately as I would like. It's confusing, I suspect for Jorjy and many of us, because we are used to acceleration changing speed,not time and yet in Einstein's spacetime acceleration acts analogous to rotational motion in space (where aceleration is perpendicular to velocity), speed remains constant while direction changes, but acceleration in spacetime also changes passage through TIME...the faster one moves the slower the passage of time for them....

The best way so far for me to "visualize" that is that linear acceleration in space appears as a curve in spacetime.....
I think maybe one thing you are not appreciating is that nothing moves in spacetime. Motion implies the passage of time, but time is one of the dimensions of spacetime. Movement of a particle through space corresponds to a static line in spacetime. The kinematics of particles in space corresponds to the geometry of static worldlines in spacetime.

The relative 3-velocity of a particle relative to an observer corresponds to the angle between the particle's worldline and the observer's worldline. In curved spacetime this can be defined unambiguously only where the two worldlines intersect. 4-acceleration is therefore a change of angle, i.e. the curvature of a worldline, i.e. the reciprocal of the radius of curvature of a worldline.

pervect
Staff Emeritus
pervect posts

Why would the notion of time of the test particle and that of the mass be the same when the gravitational fields (potential) are different? I assume you imply here the relative velocities of the mass and the test particle are the same....??
Well, the most rigorous approach is that all the tangent spaces are different, so 1 second here is never the same as 1 second anywhere else. This may be rigorous, but is more than a bit unhelpful.

The way in which I think of things is that the static geometry defines the direction of a unit of time, i.e. a second, though not its magnitude. And, given that the direction of a second is defined, the magnitude of a second can be defined by local clocks, and not by coordinates. This POV is not quite compatible with the common idea of "time dilation", however. The second I think of is always the "local" second, not the second of some particular global coordinate system.

On the other hand, coordinates are purely conventional, that one can and does use a lot of different coordinate systems, so they don't really matter. Unless one takes them too seriously, somthing that unfortunately tends to happen a lot (esp. here on PF).

I'm not sure how much this helps, but it's the best answer I have at the moment.

DrGreg Post#18
I think maybe one thing you are not appreciating is that nothing moves in spacetime. Motion implies the passage of time, but time is one of the dimensions of spacetime. Movement of a particle through space corresponds to a static line in spacetime. The kinematics of particles in space corresponds to the geometry of static worldlines in spacetime.
You are right, I do not yet "appreciate" that "nothing moves in spacetime",,,

I have the opposite view I believe. Not that I am concluding either is right or wrong, but definitely different perspectives.

The perspective I posted comes from a three dimensional plot, two of space in a flat plane and a third,time, perpendicular..... and accompanying discussion of Brian Greene's, THE FABRIC OF THE COSMOS, page 61. That happened to be the first physical/intuitive interpretation I saw and so have stuck with it.

Greene envisions that everything moves through spacetime at c, and I thought we had agreed on that earlier...now I suspect not. An acceleration acts perpendicular to velocity in spacetime and causes an object to shift from 'movement' (or passage) through time to more movement through space...so movement through time slows as a particle/test oject accelerates and speed through space increases.

Anything you can add to your post above and or a refernce that I can read online wth more details would be especially interesting....

pervect posts:

The way in which I think of things is that the static geometry defines the direction of a unit of time, i.e. a second, though not its magnitude.
How are you using the term "static" here....you don't mean that the geometry in the vicinity of the object and the test particle are the same, right?? Or are you thinking incrementally small around a point at each location which would be approximately flat...the same or "static" in that sense???

I hate to "hijack" a thread like this but I hope this discussion helps the OP as we (he and I ) seem to have a similar interpretation problem....