# How does matter accelerate in a gravitational field

1. Feb 20, 2013

### peterf1

I have been reading various forums etc but can't find a clear explanation of how the space-time warping causes an object to accelerate/gain energy.

I don't want pseudo explanations about potential energy etc.

Thanks

2. Feb 20, 2013

### WannabeNewton

Are you sure you are phrasing your question right? A massive test particle in free fall (no non - gravitational interactions) has no 4 - acceleration i.e. $\triangledown _{U}U = 0$ where $U$ is the 4 - velocity tangent to the particle's world line.

3. Feb 20, 2013

### Staff: Mentor

Hi peterf1, welcome to PF!

The reason that you haven't found a clear explanation is because it is not correct. Suppose that you have an apple which falls and hits Newton in the head. In a local inertial frame the apple doesn't accelerate at all and its energy is constant, Newton accelerates so that his head hits the apple. In an earth-fixed non-inertial frame the apple accelerates and gains KE, Newton remains at rest.

In both cases the space-time warping is the same, what is different is merely the coordinate system. Energy is not an inherent property of an object, but rather a coordinate-dependent quantity.

4. Feb 20, 2013

### peterf1

But if the apple hit newtown in the head just after it left the tree as opposed to when it is just about to hit the ground there is more energy in the collision isn't there?

5. Feb 20, 2013

### Drakkith

Staff Emeritus
Absolutely.

6. Feb 20, 2013

### Staff: Mentor

Yes. That is a coordinate independent fact that can be explained by spacetime warping. However, I will have to post that tomorrow.

7. Feb 20, 2013

### Staff: Mentor

Just after it left the tree? Meaning, say, Newton is standing on a ladder just under the apple, instead of sitting on the ground? That would mean *less* energy in the collision, because the relative velocity of Newton and the apple is smaller, therefore the kinetic energy is smaller.

8. Feb 20, 2013

### Drakkith

Staff Emeritus
Whoops, I must have read that backwards.

9. Feb 21, 2013

### SinghRP

I read Peterf1's question at Post 2. Does anyone believe this question has been answered? Can anyone answer it without using matn?

10. Feb 21, 2013

### Staff: Mentor

OK, to understand relativity, especially GR, the first thing that you need to understand is that it is fundamentally a geometrical theory. Physical things are mapped to geometrical things in the theory.

First, the position of an object as a function of time is mapped to what is called a "worldline". It is literally a geometrical line drawn in spacetime.

If the object is inertial (i.e. in free-fall so an attached accelerometer would read 0) then its worldline is straight in a coordinate independent sense. Another word for that is that the line is a "geodesic". Objects that are not inertial (i.e. not in free-fall so an attached accelerometer reads some non-0 value) have worldlines that are curved in that same coordinate independent sense. The direction of the accelerometer reading is the direction that it is curved and the greater the accelerometer reading the tighter the curve.

So, Newton's head is not in free fall and an accelerometer attached to Newton's head would indicate that the acceleration is up, therefore his head's worldline is curved upwards. The apple is in free fall so its worldline is straight. The apple hits Newton's head when their worldlines intersect. The angle that their worldlines form when they intersect gives their relative speed, the greater the angle the faster their relative speed, and therefore the more energetic the collision.

Newton's head and the apple are initially at rest wrt each other, geometrically their worldlines are initially parallel. The further apart they start the more "distance" (aka spacetime interval) that the head-worldline has to curve before it intersects the apple-worldline. So the further apart they start, the greater the curving, the greater the angle, and therefore the more energetic the collision.

I hope that helps.

EDIT: Please see the animated diagram that A.T. posted below.

Last edited: Feb 21, 2013
11. Feb 21, 2013

### A.T.

Space-time warping causes coordinate acceleration (dv/dt) in the rest frame of the surface. The proper acceleration (accelerometer) in free fall is zero in any frame. Here an animation that compares Newtons and GR gravity. Note that that in GR no force is acting in free fall, hence a straight world line and zero proper acceleration:

12. Feb 21, 2013

### peterf1

Ok I think I understand that but I don't see why the "greater the angle the more energetic the collision". Where does that energy come form?

13. Feb 21, 2013

### Staff: Mentor

Intuitively: The greater the angle, the greater the closing speed, so the more forceful the collision. Think about driving a car into a brick wall at a shallow glancing angle versus a steep head-on angle.

14. Feb 21, 2013

### Staff: Mentor

Hi guys. I'm a little confused over this video, and maybe someone can help me out. First let me say that, as an conceptual depiction for how gravity works, it is a zillion times better than the "rubber sheet analogy". But, maybe I'm missing something. It doesn't look like any curvature of spacetime is involved. It just looks like there are curvilinear coordinates, but the coordinates are in a flat spacetime. Even when it is wrapped around a cone, the spacetime still seems flat to me (i.e., a cone surface is a flat space). I'm confused about how this example should be interpreted physically.

The gravitational analogy that works well for me is presented in MTW where two observers are moving (in free fall) north on the surface of a sphere along lines of constant longitude and are getting closer together. The latitudinal direction is the analog of time. At least in this case, the spacetime is curved, although the curvature is not caused by the bodies.

15. Feb 21, 2013

### Staff: Mentor

This isn't quite right, because all of the motion is radial; there is only one relevant space dimension, so there is no variation in spatial angle, and the brick wall analogy requires a variation in spatial angle. (Also, if the two collisions were both at the same relative velocity, the damage would be about the same in either case; it would depend more on the details of the crashworthiness design of the vehicle than on the geometry of the collision.)

The angle in the case of Newton and the apple is an angle in spacetime, and it represents relative velocity: the greater the angle, the greater the closing speed, as you say. But the spatial angle is the same.

16. Feb 21, 2013

### HallsofIvy

We know that objects move in geodesics in curved "space-time". Why that is true, we don't know- any more than we knew why, in Newtonian physics, "gravity" caused all mass to attract one another.

17. Feb 21, 2013

### peterf1

This is all good but I suppose it leads me to my next question -

How do collisions work (on an atomic level)?

18. Feb 22, 2013

### A.T.

Actual curvature is not relevant locally. It causes tidal forces over an extended area. But the local effect of "gravity pull" doesn't depend on curvature. On board of an rocket accelerating in flat spacetime you have gravity too, which is locally equivalent to the gravity on Earth's surface.

The rolling of the diagram has no physical significance. It doesn't change the intrisic geometry of the space time patch. It is done solely for visualization purposes:
- the unrolled state seen initially shows the geodesic property of the falling worldline better
- the cone-like rolled state has better correspondence to the apple as we see it falling radially

That is tidal gravity. It is connected to curvature and 2nd metric derivatives. But local "gravity pull" is connected to the 1st derivative of the temporal metric component (gradient of gravitational time dilatation).

19. Feb 22, 2013

### Staff: Mentor

The relative speed is the tanh of the angle between the worldlines in units where c=1. The greater the relative speed, the higher the energy of the collision.

20. Feb 22, 2013

### DrGreg

This is my own non-animated way of looking at it:

• A. Two inertial particles, at rest relative to each other, in flat spacetime (i.e. no gravity), shown with inertial coordinates. Drawn as a red distance-time graph on a flat piece of paper with blue gridlines.
• B1. The same particles in the same flat spacetime, but shown with non-inertial coordinates. Drawn as the same distance-time graph on an identical flat piece of paper except it has different gridlines.

B2. Take the flat piece of paper depicted in B1, cut out the grid with some scissors, and wrap it round a cone. Nothing within the intrinsic geometry of the paper has changed by doing this, so B2 shows exactly the same thing as B1, just presented in a different way, showing how the red lines could be perceived as looking "curved" against a "straight" grid.
• C. Two free-falling particles, initially at rest relative to each other, in curved spacetime (i.e. with gravity), shown with non-inertial coordinates. This cannot be drawn to scale on a flat piece of paper; you have to draw it on a curved surface instead. Note how C looks rather similar to B2. This is the equivalence principle in action: if you zoomed in very close to B2 and C, you wouldn't notice any difference between them.

Note the diagrams above aren't entirely accurate because they are drawn with a locally-Euclidean geometry, when really they ought to be drawn with a locally-Lorentzian geometry. I've drawn it this way as an analogy to help visualise the concepts.

#### Attached Files:

• ###### Curved spacetime v2.png
File size:
33.8 KB
Views:
3,842
Last edited: Jan 14, 2016