# An Apparent Contradiction in General Relativity?

1. Jul 7, 2014

### Juan Largo

I'm reading a book about unification of general relativity and quantum physics. The author states that one of the axioms of GR is background-independence, meaning that space-time has no external reference points. All distances and motions in space-time are relative. On the other hand, the standard interpretation of gravity according to GR states that two objects fall toward each other because their straight-line geodesics intersect as the objects travel through curved space-time.

Every book I've read about GR gives the same explanation. However, the very notion of traveling along geodesics through curved space-time clearly violates background independence, which is the basis of GR. If all objects are traveling together "through" space-time (being background-independent), then where could they be "traveling" to? What makes them travel? What is their "destination"? Is there another way to interpret gravity within the context of GR without violating background independence?

2. Jul 7, 2014

### Staff: Mentor

Do you have a reference supporting this claimed contradiction?

I don't see how the notion of traveling along geodesics violates background independence as long as the distribution of matter is what determines the geodesics. However, I also don't see background independence as a foundational principle of GR.

3. Jul 7, 2014

### Juan Largo

Why is a reference necessary?

1. If all things travel "through" space-time, there is a preferred direction of motion.

2. Frame independence means there are no preferred directions or locations.

It seem pretty obvious that Statement 1 contradicts Statement 2.

4. Jul 7, 2014

### phinds

Because that is how this forum works. If you make a statement, you have to be able to back it up with mainstream science, not your own opinion or your interpretation of what you have read.

5. Jul 7, 2014

### UltrafastPED

So that we can look at the source of the claim, and make an independent evaluation.

6. Jul 7, 2014

### UltrafastPED

What if "things" just happen to be travelling in all possible directions?

7. Jul 7, 2014

### Staff: Mentor

https://www.physicsforums.com/blog.php?b=2703 [Broken]

It's hard to be really precise with natural language; when people speak of a "preferred direction" they nearly always mean "preferred spatial direction". Time is different from space even in theories such as GR that treat both as directions in a four-dimensional spacetime.

Last edited by a moderator: May 6, 2017
8. Jul 7, 2014

### Staff: Mentor

Perhaps a better definition of background independence will help:

http://en.wikipedia.org/wiki/Background_independence

I think the OP is thinking that there is no coordinate system when dealing with two bodies alone in space so how can there be a geodesic.

However in this case, an observer could measure the distance between them as one coordinate and use his/her timetime as a second. That would constitute a spacetime coordinate system from which a geodesic could be discerned.

http://en.wikipedia.org/wiki/Geodesics_in_general_relativity

9. Jul 7, 2014

### MikeGomez

So is this puzzle equivalent to "Why does time flow in one direction?" ?

10. Jul 7, 2014

### Juan Largo

Good-bye then.

11. Jul 7, 2014

### Staff: Mentor

Please reconsider, many of us have gone through this phase when we comment on threads and see it vanish because we violated some PF rule, or have been chastised when we give some student an answer to a problem when the student shows no effort.

We deal in mainstream science here with the goal of helping students better understand what they are learning. To discuss speculative things or personal theories confuses some students and muddies the water to understanding.

So read some more posts, read the PF rules and then decide if you really want to quit. Compare PF with other forums and you'll see that this is a nice community where discussion in the pursuit of learning is promoted and where you'll will find better quality information on a particular topic.

12. Jul 7, 2014

### stevendaryl

Staff Emeritus
It often helps in understanding the geometry of spacetime to look for analogies with something you already understand. A simple analogy for spacetime is a blank piece of paper. An analogy for the path of a particle is a continuous curve drawn on that piece of paper. You put down a pencil at some point $P_0$, and you start drawing, without picking up your pencil (and without crossing itself). Note that such a curve has a preferred direction, which is the direction away from the starting point, $P_0$.

The piece of paper has no preferred direction, but the curve does. The curve also has a notion of "progress". Any point $P$ along the curve can be associated with a real number, $s$ which increases as you get farther and farther away from $P_0$. The simplest choice is to let $s$ be the distance along the curve from $P_0$, but you could use any other real-valued parameter, as long as it keeps increasing as you move along the curve.

Now, once we've parametrized the curve by a progress parameter $s$, we can re-interpret the curve as a point object "traveling" across the page. Travel in this sense just means that you have a real-valued parameter $s$ that is increasing (it's the curve's notion of "time passing"), and you have a position $P(s)$ that changes as a function of $s$. You can also define a notion of the "velocity" of the object. In terms of coordinates, the velocity is the pair $(\frac{dx}{ds}, \frac{dy}{ds})$, where $(x(s), y(s))$ is the coordinates of the curve at parameter $s$.

So you definitely don't need a preferred direction in order to talk about "traveling". All you need is a way to parametrize paths. In GR, the path taken by a massive object can be parametrized by "proper time", which you can think of as the time shown on a clock that is attached to that object.

13. Jul 7, 2014

### bhobba

That applies only spatially. For time we have the principle of homogeneity which says all instances of time are equivalent - not directions.

And it is what defines an inertial frame - coordinate systems in GR are a lot more general than that.

Thanks
Bill

14. Jul 7, 2014

### phinds

I take it you don't plan on a career in science or engineering. If you do, you might want to reconsider, given your attitude.

15. Jul 7, 2014

### Staff: Mentor

Which book? As others have pointed out, it helps to give specific references. It's particularly helpful when it seems likely that you have misinterpreted what you read; see below.

Can you give a direct quote? The last part of that sentence ("space-time has no external reference points") is not what background independence means, but I can't tell whether that's your misinterpretation or the author's.

Background independence in GR means that the entire spacetime geometry is dynamic, i.e., there is no "fixed background" part of spacetime that is not included in the Einstein Field Equation, which is the dynamical equation linking spacetime geometry to its "source", matter and energy.

This is true, but it's not what "background independence" means. All distances and motions are relative in special relativity too, but SR is not background independent: the geometry of spacetime is fixed in SR, not dynamic.

Yes.

No, it doesn't. See above.

To other regions of spacetime. Spacetime is the entire universe in GR; there is nothing "outside" it. But there doesn't have to be to be a place for objects traveling on geodesics to "travel to"; they just keep traveling to other parts of spacetime.

However, all this depends in any case on this "traveling" interpretation, which is just an interpretation and is not a necessary part of the physics. You can describe everything in GR in terms of 4-dimensional spacetime, and all the geodesics within it, just "existing", without anything having to "travel" at all. The "traveling" interpretation is a conceptual crutch we humans use because it makes it easier to link up the physics with our ordinary experience.

Nothing. "Traveling" is an interpretation, not physics. See above.

The "traveling" interpretation has nothing to do with background independence; you can put the same interpretation on SR, which is not background independent, as I showed above. GR is background independent for the reason I gave above.

16. Jul 7, 2014

### atyy

Can one have exact geodesic motion without violating background independence?

When the backreaction is included, I don't think a small body moves on an exact geodesic. For example, "up to order ε2 errors, a point mass m moves on a geodesic" http://arxiv.org/abs/1102.0529 (p144)

17. Jul 7, 2014

### pervect

Staff Emeritus
It's probably too late in this individual case, but I'd like to clarify something. Debating the correctness of relativity (or other major scientific theories) is outside the scope of physics forums.

Titles like "an apparent contradiction" in relativity (or whatever) such as in this thread don't tend to get much sympathy, it's presumed that the person is looking for a debate, as opposed to answers.

What we can do is try to explain and teach the theories to those who are interested in how they work - this is different than debating them. If one is interested in debate, there are other forums for such "debates", though I can't say how much luck one will have finding technically qualified people who are interested in debating, especially debating with people who aren't technically qualified. Usually the people who debate are the people who like debating as an art into itself, the technical facts wind up subordinate to the desire to "win the debate". In my experience the quality of such "debates" turns out not to be very high, and it also turns out in the end to not be an especially good way to learn the ins and outs of a highly complex subject like relativity (or whatever is under discussion).

While I personally dislike debate and find it mostly unproductive, the forum guidelines don't prohibit it. They do attempt to minimize the worst sort of debate, "uninformed debate" by requiring that the topic under debate have some reference to the published literature, and that the arguments be of a professional level with references to said literature. That's not necessarily a guarantee that the people who are debating actually know anything, but it helps :)

This is my take on the forum guidelines https://www.physicsforums.com/showthread.php?t=414380, which are ultimately interpreted by the staff here.

18. Jul 7, 2014

### Staff: Mentor

In other words, you're saying that exact geodesic motion presupposes no back reaction, and no back reaction means background dependence--the spacetime geometry is not allowed to respond to the small body's stress-energy. I think this is true, but I think the answer to it is that GR does not require exact geodesic motion except for a test body, which by hypothesis has no influence on the spacetime geometry. In other words, exact geodesic motion is an idealized limiting case, and GR does not claim it to be anything more than that. The question of how close the actual motion of actual bodies is to exact geodesic motion is left open in GR; it depends on the specific situation.

19. Jul 7, 2014

### Staff: Mentor

I would add that saying "I read somewhere that...", or words to that effect, followed by something which is clearly a misunderstanding of relativity, without giving a reference, also doesn't tend to get much sympathy. I think a lot of posters who do this aren't "looking for a debate, as opposed to answers"; they just don't realize how much they've misunderstood what they've already read, so they don't understand how much they're going to have to change their conceptual scheme in order to understand how relativity actually works.

Of course, combining this with a title such as you describe only makes it worse.

20. Jul 7, 2014

### atyy

Yes. So there is, I think, in spite of the OP's misunderstanding of what "background independence" means, some part of the OP's question which is in fact a good question, since one does hear:

1) GR is background independent.
2) Test particles travel on geodesics in GR.

If one interprets "test particles" as small bodies, eg. in the usual presentation of the perihelion precession of mercury in the Schwarzschild background, then (1) and (2) are contradictory.