# Rulers on a heated slab

1. Jun 16, 2006

### pervect

Staff Emeritus
Let me start with a longish quote from Einstein

http://www.bartleby.com/173/24.html

In the past, I have characterized this as a matter of "coordinates", but upon thinking about this, that's not quite right - the word choice is incorrect. (I think I'm expressing myself better this time around, but there may still be room for technical improvement).

It is really upon introducing a metric (and not coordinates) that the geometry of the disk becomes non-Euclidean. We can assign coordinates to the marble surface however we like, and a zero Riemann tensor will remain zero. When we change the metric, however (our defintion of distance), we _can_ turn a flat marble surface into a non-flat one.

Last edited: Jun 16, 2006
2. Jun 23, 2006

### actionintegral

In layman's terms, explain your use of the word "flat" here. What concept does it describe in this situation?

3. Jun 23, 2006

### Garth

Euclidean geometry.

EDIT for clarity
If the outer edge of the marble slab itself were heated then the surface would become hyperbolic, if frozen then spherical.

If the marble slab remained flat but the outer rulers were heated then the surface would appear spherical, if frozen then hyperbolic.

i.e. The interior angles of a triangle sum to 1800,
if the surface were hyperbolic they would sum to < 1800,
is the surface were spherical they would sum to > 1800.

But note that the angle subtended at the centre by two distant points would not be affected, this deformation of the coordinate system is a conformal transformation. It would be the other two interior angles that change with a change in surface curvature.

Likewise on a flat surface
the circumference of a circle would be 2$\pi$r,
if the surface were hyperbolic
the circumference of a circle would be > 2$\pi$r,
if the surface were spherical
the circumference of a circle would be < 2$\pi$r.

All as measured by the little rod-ruler system centred on the observer at the centre of the disk.

I hope this helps, it is an important concept in conformal gravity theories.

Garth

Last edited: Jun 24, 2006
4. Jun 23, 2006

### actionintegral

So the marble table looks "as if" it were a sphere, or it looks "indistinguishable" from a sphere. "Flat" and "non-flat" are metaphors.
I see.

What's the common term for "non-flat"?

5. Jun 23, 2006

### Garth

Yes, from the centre the possible geometry of the outer regions cannot be distinguished from possible changes in scale.

In cosmology this means that in order to make a statement such as "The WMAP data is consistent with a spatially flat universe" an assumption has also to be made: "Rulers maintain a fixed length, i.e. atoms do not change size, over cosmological time scales."

A more comprehensive statement about the WMAP data would be to say that it is consistent with a conformally spatially flat universe.

One reason why rulers may not maintain a fixed length is if atomic inertial masses, or Planck's constant, or the speed of light, vary over such time scales. Various alternative theories explore each of these possibilities.

Non-Euclidean.

I hope this helps.

Garth

Last edited: Jun 23, 2006
6. Jun 23, 2006

### Robert100

I think I appreciate Einstein's analogy, but I am uncertain of its purpose. Is he saying that spacetime really isn't curved, but it appears that way because of the effect that spacetime has on any physical method we use to make measurements with?

The way I was taught, Einstein claimed that spacetime itself was curved. This curvature led to a change in the motion of bodies, this creating the illusion of a gravitational "force". Also, (continuing what I originally learned) General Relativity's model of gravity has no need or use for gravitons or virtual gravitons. Objects just attempt to move in accord with their inertia, but this curved space influences their actual path. In a sense, gravity isn't a force at all, unlike the other three forces.

However, this quote you bring forth disturbs me, because it sounds as if Einstein is saying that spacetime really isn't curved. Rather, spacetime in some way is not uniform, and this non-uniformity affects the way that we measure things. Thus, spacetime only appears to be curved, but it really isn't. Am I understanding you and he correctly so far?

Or are you saying that we simply do not know if spacetime is curved, and your analogy is showing us that it may be difficult (if not impossible) to discern between a curved spacetime, and a spacetime that only appears to be curved?

If you mean the former, then in what way is spacetime not uniform, such that it affects all our physical measurements? (I had previously thought that it was not uniform in geometry, but Einstein's quote and your discussion seem to imply that this isn't true. Your analogy implies that it is non uniform in way that affects our measurements to give the illusion of curvature!)

Can you help clarify?

Robert

7. Jun 23, 2006

### pervect

Staff Emeritus
Let me start out by saying that when you use standard rulers (and standard clocks) space-time is curved. This is unambiguously true, but it does depend on using existing standards.

I'm also talking a bit about the nature of curvature (more precisely, extrinsic curvature). To define extrinsic curvature mathematically, one needs a way of measuring distances. This is done in relativity via a metric.

2-d manifolds make things very simple, so I am really talking about curvature on a 2-d manifold (i.e. a plane).

A flat (non-curved plane) can be completely tesselated with squares. Squares in this context are defined as quadrilaterals with 4 equal sides, and 2 equal diagonals (the diagonals are equal to each other, not equal to the sides).

We know how to build squares as long as we are able to measure distances - thus having a ruler alows us to build squares.

It is intsturctive to think about what happens when we attempt to tesselate a sphere with squares. We find that it does not work. Spheres have a positive extrinsic curvature. (In 2 dimesions curvature is especially simple - it's a single number. This is not true in general, in higher dimensions curvature needs a lot of numbers arranged in a tensor to describe it).

In order to cover the surface of a sphere, we can't use squares. Without getting too technical, I hope I can say that we find that the problem is that there is an "excess of material" when we attempt to cover the surface of a sphere with squares. We need to make some of our squares smaller by making the rods shorter to get proper coverage.

The opposite hapens if we attempt to cover a saddle surface. We need to make some of our rods longer to cover such a surface.

While I did not explicitly say so in my previous post, it turns out that it is possible, by using non-standard rulers and clocks to make space-time appear perfectly flat. This is not the usual approach to relativity, but an alternate way of understanding relativity that is essentially equivalent to the usual way.

This is published in, for example,

http://xxx.lanl.gov/abs/astro-ph/0006423

I am not forcefully advocating this point of view, but it is wortwhile I think to point out that gravity can be understood in this particular manner.

I have not gone into great detail of the other consequences of using non-standard rulers and clocks. I will just say that it is not a trivial effort, many formulations of physical laws are based on assuming that standards are adhered to and will not be true as written if the standards are not adhered to.

I will say that one has has the interesting (and totally philosophical) choice between a static, flat space-time and dynamical, changing rulers, or one can have a dynamnical, non-flat space-time and static rulers. The usual formulation of relativity takes the later course.

When one talks about "reality", ie

this should be a clue that one is talking not about physics, but philosophy. This may serve as an example of how two apparently different approaches taking different philosphical views are experimentally equivalent.

8. Jun 24, 2006

### Garth

See my edit for clarity in post#3
In the one case it is the surface itself that becomes curved, in the other it only appears that way because the rulers vary in length. As we can only observe the distant universe from afar we cannot tell which is actually case for the universe.

Then the two alternative views have to be tested by internal and experimental consistency. As has been said if you allow rulers and clocks: matomic, h, c, or $\alpha$ to vary there are other serious consequences.

There may, of course, be serious consequences in not allowing them to vary, such as might be indicated at present by the standard model requiring inflation, non-baryonic DM or unknown DE while they remain undiscovered in laboratory physics.

How do we remotely measure objects at distance? What is the Ground-Truth by which we can verify the conclusions?

It is best to keep an open mind on these questions.

Garth

Last edited: Jun 24, 2006
9. Jun 24, 2006

### Robert100

I had asked "Is Einstein saying that spacetime really isn't curved, but it only appears that way?"

Pervect replied: "this should be a clue that one is talking not about physics, but philosophy. This may serve as an example of how two apparently different approaches taking different philosphical views are experimentally equivalent."

I disagree; how is this philosophical? Are you using the term "philosophical" to mean "We don't currently know how to make this measurement?"

Consider two people standing on the Earth's equator. They measure their distance apart as 1 km. Both then decide to walk towards the geographical north pole, at the same speed. As they move further north, each sees himself as following a straight line. Yet curiously they find that the distance between them keeps shrinking. When they arrive at the north pole, the distance between them is zero (they bump into each other!)

When one asks "Is the Earth a sphere, or is it really flat and some genuine force attracts them", is that a philosophical question? No!

We have learned that while the Earth appears to be flat, from our local perspective, the Earth really is a three dimensional sphere. (An oblate spheroid, but that's not relevant to our discussion.) As the two people walk north, it appears from their naive, ground-based perspective that some mysterious "force" pulled them together.

But we who see the Earth from a more accurate perspective know the true story: The world is not flat, and no force attracts the people. The apparent force is more of a fictional force.

That's the same question I am asking about space. From our point of view, space locally appears "flat". (Flat, in a 3-D sense. What is the word for this? Is there a word for 3D Euclidean volumes?) Yet ever since Einstein alerted us to this possibility, we have looked for clues that our own space is curved or warped. We found these signs.

So my question is this: Although men on the ground don't see the real, actual curvature of Earth in space, the Earth actually is curved. (The curvature can't be seen from the ground, but its existence can be inferred from ground-based measurements and Occam's razor.)

I am asking if the same is true of space itself. Is space really curved by the presence of massive bodies? Perhaps we have no easy way of measuring this from within our spacetime, but I am asking if this is a question which can be answered in principle, even if we currently may not have a good way of measuring this.

And if space is not really curved, then what is causing the "change in rulers"?

I know that we can play math games to make a space of any curvature look flat; we use an assortment of different length rulers, etc. We can also play math games to make flat space looked curved. But this kind of math game can go on indefinitely. We can come up with an infinite number of measurement schemes in which any space can be mapped into any other space. But surely physical reality means something, surely Occam's razor comes into play.

Robert

10. Jun 24, 2006

### pervect

Staff Emeritus
Let's try the more mathematical approach.

Suppose we have a manifold, but we don't have a metric. Without a metric we cannot compute the curvature tensor. We can't say that the manifold is curved, or not curved.

There are a few things that we can do at this point, with various topological invariants, so we can talk about connectedness, dimension, winding number, the inside and outside of a curve, etc. But in order to talk about curvature, we need to introduce a metric.

Topology isn't my strongest point, but I'll give a brief quote to show the sorts of things we can and can't talk about:

http://planetmath.org/encyclopedia/TopologicalInvariant.html

Thus we can do all of the above things without a metric. Euler characteristic is probably familiar as "vertices - edges + faces". Homology groups, are related to counting holes in the topology.

We can't, however, manage to compute curvature without the extra information in the metric. As I look at the math, specifically the Gauss-Bonnet theorem, there may be some deep water here - maybe some of our more mathematical types can comment more.

Now, let's consider your example:

By talking about the distance between two points, you've just introduced a metric - a notion of distance.

When we use standard rulers to define our metric (our notion of distance), we can say that the surface of the Earth is definitely curved, and we can also say that space-time is definitely curved.

Since using standard rulers is very much to be desired, I would suggest that thinking of them as being "real rulers" isn't that bad of an idea. Thus if I say "standard ruler" and you think "real ruler", and I say "non-standard ruler " and you think "some crazy mathematical abstraction", I think we'll be communicating reasonably well.

Last edited: Jun 24, 2006
11. Jun 27, 2006

### pervect

Staff Emeritus
On the mathematical side, I'd like to stick my neck out a bit, and throw out the following remark.

Perhaps I can get Hurkyl or some other of our more mathematically inclined readers to comment.

When I say that we can't define curvature without a metric, this really applies to the local topology.

Global aspects such as the Euler characteristic can tell us that we don't have a globally Euclidean geometry, for instance.

We still can't define the curvature tensor at the neighborhood of a point without a metric. AFAIK, anyway.

This actually turns out to be an issue with the approach I mentioned earlier (the approach using non-standard rulers in a static Minkowski space-time) in terms of the global topology.

(This is text from Chris Hillman, though it appears on Baez's webpage).

12. Jul 20, 2006

### MeJennifer

Do you mean that space-time is observed as being curved by standard rulers and standard clocks?

Perhaps I am wrong, but it seems to me that the type of rulers could not depend on whether space-time is flat or not.

13. Jul 20, 2006

### pervect

Staff Emeritus
If you like, you can put "observed" in my statement. The ruler example shows that one set of rulers finds that the slab is curved, while another set of rulers finds that the slab is flat. That's the important point of the example, in my opinion.

The two points I want to make:

rulers are necessary to define local curvature. Without them, you can't measure curvature at all.

different rulers may give different answers as to whether or not a particular space is "curved" or "flat".