# Straight line through spacetime.

1. May 10, 2010

### Rotormaster

Since to travel in a straight line through space time requires the subject in question to have zero change to its state of motion, ie maintain constant velocity at all times. Does this mean that the only entity to ever have and that ever will achieve a straight line through space time is light?

2. May 10, 2010

### Passionflower

The path length of a photon in spacetime is zero, it is a point.

3. May 10, 2010

### Fredrik

Staff Emeritus
Any particle can move as described by a "straight line" in spacetime. The technical term is "geodesic". There are three different types of geodesics: Timelike, null and spacelike. A timelike gedesic represents the motion of a massive particle in free fall. A null geodesic represents the motion of a massless particle. (Massless particles are always in free fall if they're moving through a vacuum).

If you meant to ask specifically about null geodesics, then the answer is that any massless particle in free fall would move as described by a null geodesic. The only massless particles in the standard models are photons and gluons, and I don't think gluons are ever in a state that can be described as free fall. So I guess photons are the only particles that will move this way, at least over long distances. But there could exist other (massless) particles that do the same (if they are very rare or interact very weakly with other kinds of matter...otherwise they would have been discovered by now).

4. May 10, 2010

### Rotormaster

But aren't photons the only particle with constant velocity? would this not mean that since everything started they would be the only thing to maintain absolute synchronization with time's arrow ie achieve a straight line through space time?

5. May 10, 2010

### Ich

Moving "in a straight line" in curved spacetime means moving "along" a geodesic. From the moving particles' point of view, there is no change of their "state of motion" as they are alwasy at rest wrt themselves and feel no forces acting on them. (Photons don't have a point of view, btw.)
Measured in some coordinate system, their speed and direction may change. That's true for light also.

6. May 10, 2010

### Fredrik

Staff Emeritus
No. Einstein's equation tells us which curves in spacetime are geodesics. We then define zero acceleration as "motion described by a geodesic", and acceleration as a specific measure of much a particle's motion deviates from geodesic motion. So if there are any timelike geodesics at all in spacetime (and there always are), then massive particles can have zero acceleration as well.

"Constant velocity" isn't a very precise term. We can make the velocity anything we want it to be, and depend on the time coordinate in any way we want it to, just by choosing the coordinate system carefully. That's why I'm talking about zero acceleration instead. This is a coordinate-independent concept.

7. May 10, 2010

### Staff: Mentor

Careful here. The Minkowski norm is degenerate, so the fact that two points have a separation of 0 does not imply that they are the same point like it would with the Euclidean norm.

8. May 13, 2010

### stevmg

Is your statement due to the so-called "null" line or surface of the space-time coordinate system? (In other words, if something is travelling at c or light speed, wouldn't the interval be $$\Delta$$x2/c2 - $$\Delta$$t2 = 0) Neither spacelike nor timelike.

Am I on the right track here?

Remember, I am one of those uneducated ones...

9. May 13, 2010

### stevmg

Can anyone out there explain the difference between "Aristotilean" and "Galilean" spacetime viewpoints? They look pretty similar to me.

10. May 13, 2010

### Staff: Mentor

Yes. In Euclidean geometry the set of all points with 0 distance from the origin is a single point, but in Minkowski geometry the set of all points with 0 interval from the origin is a cone called a light cone.

11. May 13, 2010

### Passionflower

However, the shortest possible distance between any two points inside the cone is also 0, it is only when we restrict photons to go only forward in time that the points are only on the cone.

12. May 13, 2010

### stevmg

Help...

13. May 13, 2010

### matheinste

There is a lengthy description of both viewpoints in the early chapters of Geroch's General Relativity from A to B.

Matheinste.

14. May 14, 2010

### stevmg

Thanks Matheinste, I read Geroch's book and that's why I asked the question. All I know is there are straight lines up, bevelled decks of cards, etc., etc. I don't know if I am coming or going.

Maybe, if I phrase the question this way, at slow non-relativistic speeds as we are in the "real world" what is the most utilitarian approach? Forget about$$\sqrt{(1 - v^2/c^2)}$$

15. May 14, 2010

### Staff: Mentor

None of that statement is true in general. For two arbitrary points inside or even on the light cone the interval may be timelike, spacelike, or null. Also, the interval between two events is purely geometric and has nothing to do with photons nor the arrow of time.

16. May 14, 2010

### Passionflower

Sorry but you are wrong, provided we do not restrict photons to go only forward in time we can always "zig zag" inside the cone to make the distance 0.

17. May 14, 2010

### matheinste

If we are discussing the spacetime interval why is it not true that a zig zag path can give a null value if we sum along the path, even if we restrict the motion of the photon to forward pointing in time.

Matheinste.

18. May 14, 2010

### Passionflower

If we restrict it we could obviously not "zig zag" from a future to a passed event inside the light cone.

19. May 14, 2010

### matheinste

If we are referring to any two events in any time order that may be true. But we can still have a zig zag null path going forward in time between events joined by a timelilke vector within a light cone without either part of the zig zag path being on the light cone.

Matheinste.

20. May 14, 2010

### matheinste

I think that in the Aristotlean view there is a priviledged state of absolute rest. In the Galilean view there is no such absolute state, all motion is relative.

Matheinste.