# Time dilation and curved space

1. Jun 25, 2014

### Warp

I am trying to get an understanding of general relativity one tidbit at a time. I have a vague concept of why curved spacetime causes the effect we call gravity. However, there's an aspect of it (ok, there' are quite many aspects of it, but I'm concentrating on this one right now) that I can't understand.

Time passes more slowly closer to a mass. In other words, the more curved spacetime is, the slower time passes.

This seems counter-intuitive to me. On the surface of the Earth spacetime is more curved than far away from Earth. Thus it would seem that you traverse a larger distance in spacetime when you move in the time axis on the surface than far away from Earth. Yet, even though you travel a larger distance, time passes more slowly, not faster.

Could someone explain in simple and intuitive terms what's going on? (I'm probably either way off, or am simply misunderstanding a simple thing...)

2. Jun 25, 2014

### phinds

I'm not sure I have this right because I don't know the math but I think the problem is that you are using the normal English language term "curved" to describe something that really can only be properly described mathematically. The "curvature" is very real in terms of the bending of light as it goes around a massive object such as a star or a galaxy but I'm not sure you can carry that analogy over to time dilation.

3. Jun 25, 2014

### Bill_K

Yes, but it's not a direct relationship. There's derivatives involved.

1) Time dilation depends on the local value of g00, and for weak gravitational fields this is like the Newtonian gravitational potential. For Schwarzschild it goes like GM/r.

2) The bending of geodesics (particle orbits and light deflection) depend on the derivative of g00. It happens because the time dilation varies from place to place. It goes like 1/r2, and is analogous to the Newtonian gravitational force.

3) Actually, the spacetime curvature is neither of these. It's the Riemann tensor. It involves one more derivative, and for Schwarzschild goes like 1/r3.

4. Jun 25, 2014

### pervect

Staff Emeritus
There is not a linear relation between curvature and time dilation. For instance, if you drill a hole inside the Earth, at the very center there is no gravity, no curvature, and a maximum of time dilation.

So basically your whole idea of what GR is about isn't going to allow you to actually reach correct conclusions, even qualitatively. Sorry to bear the bad news.

If you are familiar with both special relativity (I'm guessing you may not be, that's my default assumption nowadays), AND the concept of geodesics, you might get something from Marolf's "Spacetime Embedding Diagrams for Black Holes". http://arxiv.org/abs/gr-qc/9806123 . It's not a particularly easy read, even if you do have the prerequisites.

If you're not familiar with special relativity, my main recommendation would be to learn it first before you try to tackle GR. You'll need to be able to understand, in particular, space-time diagrams on flat pieces of paper in the flat space-time of SR, before the idea of how to draw space-time diagrams correctly on a curved surface in the curved space-time of GR will do you any good at all.

The basic idea, though, is that GR involves drawing space-time diagrams on curved pieces of paper (rather than flat), and that the notion of "straight lines" is changed to the notions of geodesics.

If you lack either SR or the concept of geodesics, this idea may not be helpful :(.

What Marolf does in his paper is giving a Lorentzian embedding of the r-t plane of a kruskal black hole, i.e. it describes the curved shape that you'd have to draw a space-time diagram in order for all the proper times and proper distances (or in general Lorentz intervals) to have the correct lengths. It's rather similar to the idea of finding the shape of a rowboat by driving a bunch of nails into it, measuring the distance between each pair of nails, and demanding that they come out correctly.

I'll paste my favorite diagram on this point, from the "Curving" chapter of "Exploring Black Holes", downloadable from EF Taylor's website http://www.eftaylor.com/download.html - a good site with some partial introductory textbook downloads for both SR and GR. (You can't get the full textbook, just sample chapters of an old version of the textbook).

5. Jun 25, 2014

### A.T.

6. Jun 26, 2014

### Warp

7. Jun 26, 2014

### A.T.

That is correct. The bottom figure "Curved Space" just shows how the spatial geometry affects the orbit.

The top picture "Curved Time" shows what happens locally, and why free falling objects undergo coordinate acceleration towards the mass, by moving straight ahead (geodesically) in space-time. This local coordinate acceleration towards the mass results in a curved spatial path and an orbit. Here it is again animated:

It is not possible to visualize both effects in one embedding diagram, because you need a non-Euclidean 3D grid ( at least 2 space + 1 time dimension), which would require more than 3D for the Euclidean embedding space of the illustration.

8. Jun 26, 2014

### pervect

Staff Emeritus
A few more comments: The exact details of curvature in higher dimensional space are highly mathematical, but we can get a lot of leverage on understanding some general features because we live on a curved surface, the surface of the Earth. Thus we have developed (hopefully) some intuition about curvature. So lets talk about the curved surface of the Earth and how we deal with it.

There are a couple of approaches to this. One is to use a globe. This approach generalizes to using an embedding diagram. This is a good intuitive technique, but it is ultimately rather limited. Since the highest dimensional object that is easy to visualize is 3 dimensional, that means we can only handle curvature of 2d surfaces. This can still be handy, we can look at some particular 2d surface of interest, but it can't generalize to handle problems of higher dimension. Also, it's not necessarily true that we could find a 5-dimensional object whose surface represents some given 4-d space time geometry. We might need more than 5 dimensions. So the embedding diagram is a good conceptual tool, but it's limited in the number of dimensions it can handle.

The other approach is to use a metric. When we look at a map of the Earth drawn on a flat piece of paper, we know that the shapes get distorted, and distances aren't correct. With a metric, we basically handle curvature by drawing our maps on flat paper, and treating mathematically the resulting distortions we get in distances and angles.

Now, lets see what we can tell about "time dilation" using the surface of the Earth. Well, there isn't any time on the surface of the Earth the equivalent effect would be "distance dilation" or "distance contraction". One doesn't hear these terms much, and for good reason - they don't really describe what's going on very well. But lets see what they would describe if we actually used them :-).

If you consider latitude and longitude coordinates on the surface of the Earth, you can see (hopefully) that one degree of longitude represents different distances on the surface of the Earth. So the higher you go in latitude, a degree of longitude represents less and less distance. This also ties in with the idea of a metric, we can express distances on the surface of the Earth using the formalism of the metric, and certain coefficients of this metric will shrink at higher latitudes as necessary to cause the effect I've described about degrees of longitude representing lower and lower distances. I won't go into more details, if the idea isn't familiar it will probably need more words than I can spare and be too distracting.

Now, if we confusingly called degrees of longitude some sort of "distance", we could say that "distance" contracted as we went to high latitudes near the poles. Of course, we don't do that on the Earth, we realize that distance is something that's invariant to the observer, and that degrees of latitude and longitude are just coordinates. In fact we probably would find the idea of "distance dilation" as a very odd way of describing what's going on.

If we uniformly applied these ideas to space-time, we would say that proper time is the fundamental thing that's invariant and independent of the observer, and it doesn't contract, expand, or dilate, ever. But using the analogy of the surface of the Earth, we could understand how the coordinates that describe time could be said to contract or dilate in the presence of curvature.

What turns out to be important in understanding these issues is to draw a sharp and rigid distinction between the sort of time kept by clocks (which is given the technical name proper time), and the sort of time kept by coordinates (which is called by coordinate time).

The ratio of the two is what defines time dilation, I suspect what happens is that the distinction between the two different sorts of time isn't appreciated, and the whole issue gets terribly muddled as two important, but distinct, concepts get conflated - intermixed , combined, and confused.

I would also like to put a brief plug in for the philosophical idea of regrading the time that clocks keep as "real", based on the fact that all observers agree on this, and the sort of time that coordinates take as having lesser physical significance, because coordinates are just labels that we stick on a map, they are statements about the map rather than direct statements about what we might call "reality". However, I don't want to overdo this - it makes things ever so much simpler in my opinion, but ultimately it's not productive to spend too much time on philosophy.