# B Spacetime as geometry

1. Dec 6, 2017

### laymanB

I still have some confusion of the concept of spacetime as geometry. Specifically, what confuses me is causality related to this geometry. My understanding is that stress-energy of matter/energy curves spacetime, the curvature of spacetime tells matter/energy how to move, and dictates to particles the path they will take. Is this understanding correct?

1. Can we then talk about spacetime as pure geometry, as it seems to produce true effects?

2. How do you produce curvature in a mathematical object that effects causality of ordinary matter?

2. Dec 6, 2017

### PeroK

To get a proper insight into this you really have to be able to study the material to some extent. GR does has several major prerequisites. Not least, a thorough understanding of SR. Lagrangian Mechanics and a familiarity at least the concepts of vector calculus and differential geometry.

Otherwise, you more or less just have to go by soundbites of how things work.

One key factor in GR is that in the absence of forces you need a reason for a particle to do something. In Newtonian physics, changes in motion are caused by forces. In GR, you need an alternative - which is provided by the Lagrangian concept of a key quantity being maximised or minimised. In GR, a particle moves in order to maximise the proper time it experiences. This is where, in my opinion, the rubber sheet analogy breaks down, as it directly appeals to the notion of a force causing the motion.

Also, in fact, you can recreate Newton's law of gravity geometrically by:

a) Specifying a geometry for spacetime that depends on the gravitational potential.

b) Postulate that particles move in this geometry such as to maximise the proper time they experience.

From this, you can produce Newton's law of gravitation.

This is covered, for example, in Chapter 6 of Hartle's Inroduction to GR.

Now, you can read my post and digest it to some extent. But, there is no substitute for being able to follow the maths yourself and then you really understand what a soundbite like "gravity is geometry" really means.

3. Dec 6, 2017

### laymanB

@PeroK thanks for the reply. I am working my way through Taylor & Wheeler's 1966 version of Spacetime Physics. I'll probably have to reread it a few times . I have tried to watch Susskind's lectures on SR online, but I don't understand the concepts associated with the Lagrangian Mechanics, so I didn't get very far. My plan was to go onto Carroll's lecture notes on GR when I felt more comfortable with SR, but I'm not sure if they are the best resource for a beginner. Are they a good resource or is there a more basic starting book/textbook/resource, like Hartle's book you mention, that you or someone else could recommend?

4. Dec 6, 2017

### kimbyd

Space-time is described as geometry mathematically. So, essentially, yes.

Einstein's equations link the geometry to what is known as the "stress-energy tensor", which describe the energy density, momentum, pressure, and twisting forces of matter. Thus the presence of matter constrains the possible geometric shapes.

Note that the notion of "producing curvature" is a bit misleading here because the General Relativity description is a description of the geometry at all points in space and time. Producing implies some kind of cause-and-effect: first matter, then curvature. It's more that the geometry of space-time is forced to be consistent with the matter in the space-time at all points in space and time. Neither came first. And just as the geometry must be consistent with the matter, the geometry also influences how the matter moves.

5. Dec 6, 2017

### laymanB

I was thinking more along the lines of statements that spacetime is geometry. To me that statement says, the map is the territory. I.e. that there is nothing physical underlying the geometric model. That language is confusing to me because it seems to imply that something that is non-physical influences how matter moves.

It's obvious that I have some misconceptions here that need to be worked out. Thanks.

6. Dec 6, 2017

### PeroK

I believe that Hartle's book is about as straightforward as you can get when it comes to GR.

But, you know, self-study gets harder once you're at the level of GR.

7. Dec 6, 2017

### laymanB

Thanks.

Or when your brain is fuzzier than you would like it to be.

Last edited: Dec 6, 2017
8. Dec 6, 2017

### kimbyd

I think that the purpose of this kind of statement is to emphasize that the universe as we know it is fundamentally mathematical in nature. When we say that space-time is geometry, we're saying that space-time has all of the properties of the mathematical structure of geometry (specifically differential geometry). When we write down equations describing how this geometry behaves, we're describing how space-time behaves.

My thoughts on this perspective are twofold:
1. The fact that the universe appears to be fundamentally mathematical is probably correct, and a good point to make. Often we think of mathematical structures (like spheres) as being idealized, abstract concepts rather than real things. What this is saying is that if you drill down reality to its most fundamental components, the mathematics is the reality. At a macro level, something like a sphere or a plane may not exist: there's no such thing as a perfectly-round object, or a perfectly-flat one with zero thickness. But the fundamental components of our universe are, very likely, described exactly by mathematical objects.
2. General Relativity almost certainly does not describe our universe at a fundamental level. It is more akin to the sphere above: it's an idealized mathematical structure that approximates reality. The general belief is that there's a more fundamental law which, when taken to the appropriate limit, makes it so that space-time behaves very much like geometry.

Still, even though I see GR as an approximation of reality, I do think that the new perspective that GR adds to our understanding of reality is likely to hold up even if we learn a more fundamental law. For example:
1. There's no absolute sense of simultaneous events.
2. Total energy isn't conserved.
3. Speed and distance of far-away objects cannot be uniquely defined.

When we do learn a more fundamental theory than General Relativity, we're likely to get a mathematical explanation as to why space-time distances behave like geometry and why matter interacts with that geometry in the way that it does.

9. Dec 6, 2017

### Mordred

I would really follow PeroK's excellent advise on the need to look into getting a good vector calculus textbook. It is an incredible eye opener of how spacetime is described in terms of geometry. You will literally be amazed at how simple GR becomes with the mathematical tools in place and just how much of the terminology revolves around the mathematical terminology.

10. Dec 8, 2017

### laymanB

Thanks for the responses.

Let me use the analogy of two travelers starting from the equator, and separated by some angle of longitude, walking due north. We can measured the rate of change of the displacement between them and we will see that they come to the same location when they reach the North Pole. We can say that the travelers are particles and the relative acceleration between them can be explained by the geometry of the curved surface of the Earth. We have thereby just given a geometric explanation as to why the geodesic worldlines intersected, even though they started out parallel. But the curvature that explains our results is the curvature of something physical, namely an approximate sphere made of ordinary matter that we call the Earth.

So I guess my question narrows down to whether GR has any definitions about the structure of spacetime? What is it exactly that is being curved? Does GR have anything to say on this matter or is it just assumed that spacetime has a "structure" that can be curved and let's get on to modelling it so that it can be understood mathematically and used to make predictions?

11. Dec 8, 2017

### PeroK

GR says things like:

$ds^2 = -(1 - \frac{2M}{r})dt^2 + (1 - \frac{2M}{r})^{-1}dr^2 + r^2(d\theta^2 + \sin^2 \theta d\phi^2)$

There's no "structure" or "fabric" being curved. There's differential geometry. That's the real deal.

The above is the Schwarzschild geometry, which represents the geometry of spacetime outside a spherical star or planet. What you need to understand it is mathematics. Analogies are a poor substitute.

Note that it is especially the dependence of $dt^2$ on $r$ that gives this geometry its key properties - for which Newton's law of gravitation is a good approximation in many cases.

12. Dec 8, 2017

### kimbyd

No, the only description is the curvature itself.

If you want to get into the math a bit more in detail, the mathematical object is called a "manifold" (Wikipedia link).

13. Dec 8, 2017

### PeroK

I think if the OP wants to learn about GR that might not be the best place to start!

14. Dec 8, 2017

### laymanB

I ordered Hartle’s book and should have it soon. Have to start somewhere.

15. Dec 8, 2017

### PeroK

Good luck!

16. Dec 8, 2017

### Mordred

If its the same book I'm thinking of you should find it useful. A good supplement is a decent vector Calculus book. In particular on the topic of the Kronecker and Levi-Cevita connections of vector fields with regards to freefall via Principle of equivalence and covariants in regards to tidal forces.

lol I know I know small steps first :P

Last edited: Dec 8, 2017
17. Dec 14, 2017

### laymanB

Do physicists nowadays refer to gravity as a field, like Einstein did in describing his general theory?

18. Dec 14, 2017

### Mordred

Usually. I honestly can't recall any examples where it treated as a field

19. Dec 14, 2017

### Staff: Mentor

There are basically two schools of thought on this among physicists who work with GR. One school, exemplified by MTW among textbooks, places primary emphasis on spacetime geometry. The other school, exemplified by Weinberg's textbook, places primary emphasis on the analogies between gravity as a spin-2 field and other fields.

The two viewpoints are consistent, but tend to lead in different directions, for example in searching for possible theories of quantum gravity. The field viewpoint leads to the spin-2 quantum field theory approach to quantum gravity, pursued in the 1960s and early 1970s, which discovered that the field equation for this field, in the classical limit, turns out to be the Einstein Field Equation. This QFT is not renormalizable, and the general opinion seems to be that it is best viewed as a low-energy approximation to some other more fundamental theory, which many physicists think will turn out to be string theory.

The geometric viewpoint leads to different models, like loop quantum gravity, which try to explain how spacetime geometry emerges from something more fundamental, such as spin networks. The fact that what emerges from these underlying models can also be viewed as a spin-2 effective field theory is considered a useful side effect, but not fundamental.

We won't really know to what extent either of these viewpoints is preserved in a correct theory of quantum gravity until we know what that theory is. But I think it's safe to say that, however things turn out, the classical concept of a "field", which is what Einstein was talking about when he used the term in reference to gravity (and which he used in trying to unify gravity with electromagnetism), is not going to be fundamental; it's going to be a limiting case in an appropriate approximation of whatever fundamental theory we end up with.