What causes a marble slab to become a non-Euclidean continuum?

[pervect]

Let me start with a longish quote from Einstein

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

Let us now imagine that a large number of little rods of equal length have been made, their lengths being small compared with the dimensions of the marble slab. When I say they are of equal length, I mean that one can be laid on any other without the ends overlapping. We next lay four of these little rods on the marble slab so that they constitute a quadrilateral figure (a square), the diagonals of which are equally long. To ensure the equality of the diagonals, we make use of a little testing-rod. To this square we add similar ones, each of which has one rod in common with the first. We proceed in like manner with each of these squares until finally the whole marble slab is laid out with squares. The arrangement is such, that each side of a square belongs to two squares and each corner to four squares.

It is a veritable wonder that we can carry our this business without getting into the greatest difficulties. We only need to think of the following. If at any moment three squares meet at a corner, then two sides of the fourth square are already laid, and as a consequence, the arrangement of the remaining two sides of the square is already completely determined. But I am now no longer able to adjust the quadrilateral so that its diagonals may be equal. If they are equal of their own accord, then this is an especial favour of the marble slab and of the little rods about which I can only be thankfully surprised. We must needs experience many such surprises if the construction is to be successful.

If everything has really gone smoothly, then I say that the points of the marble slab constitute a Euclidean continuum with respect to the little rod, which has been used as a “distance” (line-interval). By choosing one corner of a square as “origin,” I can characterise every other corner of a square with reference to this origin by means of two numbers. I only need state how many rods I must pass over when, starting from the origin, I proceed towards the “right” and then “upwards,” in order to arrive at the corner of the square under consideration. These two numbers are then the “ Cartesian co-ordinates” of this corner with reference to the “Cartesian co-ordinate system” which is determined by the arrangement of little rods.
By making use of the following modification of this abstract experiment, we recognise that there must also be cases in which the experiment would be unsuccessful. We shall suppose that the rods “expand” by an amount proportional to the increase of temperature. We heat the central part of the marble slab, but not the periphery, in which case two of our little rods can still be brought into coincidence at every position on the table. But our construction of squares must necessarily come into disorder during the heating, because the little rods on the central region of the table expand, whereas those on the outer part do not.

With reference to our little rods—defined as unit lengths—the marble slab is no longer a Euclidean continuum, and we are also no longer in the position of defining Cartesian co-ordinates directly with their aid, since the above construction can no longer be carried out. But since there are other things which are not influenced in a similar manner to the little rods (or perhaps not at all) by the temperature of the table, it is possible quite naturally to maintain the point of view that the marble slab is a “Euclidean continuum.” This can be done in a satisfactory manner by making a more subtle stipulation about the measurement or the comparison of lengths.

But if rods of every kind (i.e. of every material) were to behave in the same way as regards the influence of temperature when they are on the variably heated marble slab, and if we had no other means of detecting the effect of temperature than the geometrical behaviour of our rods in experiments analogous to the one described above, then our best plan would be to assign the distance one to two points on the slab, provided that the ends of one of our rods could be made to coincide with these two points; for how else should we define the distance without our proceeding being in the highest measure grossly arbitrary? The method of Cartesian co-ordinates must then be discarded, and replaced by another which does not assume the validity of Euclidean geometry for rigid bodies. 1 The reader will notice that the situation depicted here corresponds to the one brought about by the general postulate of relativity (Section XXIII). 7

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.

 

[actionintegral]

we _can_ turn a flat marble surface into a non-flat one.

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

 

[Garth]

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

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

 

[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"?

 

[Garth]

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.

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.

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

Non-Euclidean.

I hope this helps.

Garth

 

[Robert100]

Let me start with a longish quote from Einstein...

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

Then Pervect writes:
...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.

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?

Thanks in advance,

Robert

 

[pervect]

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?

Thanks in advance,

Robert

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 dimensions 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

Is he saying that spacetime really isn't curved, but it appears that way

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.

 

[Garth]

When one talks about "reality", ie

Is he saying that spacetime really isn't curved, but it appears that way

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.

See my edit for clarity in post#3

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.

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: m_atomic, 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

 

[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

 

[pervect]

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

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

A topological invariant of a space X is a property that depends only on the topology of the space, i.e. it is shared by any topological space homeomorphic to X. Common examples include compactness, connectedness, Hausdorffness, Euler characteristic, orientability, dimension, and algebraic invariants like homology, homotopy groups, and K-theory.

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.

Properties of a space depending on an extra structure such as a metric (i.e. volume, curvature, symplectic invariants) typically are not topological invariants, though sometimes there are useful interpretations of topological invariants which seem to depend on extra information like a metric (for example, the Gauss-Bonnet theorem).

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:

Consider two people standing on the Earth's equator. They measure their distance apart as 1 km.

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.

 

[pervect]

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.http://math.ucr.edu/home/baez/RelWWW/grad.html

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

Reflections on Gravity, by Norbert Straumann (Institute for Theoretical Physics, University of Zurich). This brief paper offers a nice sketch of an approach to deriving the EFE which was advocated by Feynman. The basic idea is to start with Newtonian gravitostatics, considered to consist of the Poisson equation on -Minkowksi spacetime-, and then try to follow the model of how one passes from electrostatics to Maxwell's theory of electrodynamics (which is Lorentz covariant) and then to quantum electrodynamics, fixing up the approach as needed. In particular, it turns out that one must introduce back reaction of the gravitational field on matter, which leads a kind of infinite series of approximations, which was cleverly "summed" by Deser. The end result is the EFE! However, the original metric of flat spacetime turns out to be unobservable and the original hypothesis of Lorentz covariance becomes moot! Caution!: Straumann inexplicably fails to mention the fact that the approach he is discussing only yields a "local mimic" of gtr; unless one carries the "geometrization" one step further by interpreting the quantum fields as existing on one of many coordinate charts, one excludes all the solutions to the EFE which have nontrivial topology. The following paper covers the ideas Straumann is discussing from a somewhat different persepective (among many other topics).

 

[MeJennifer]

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.

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.

 

[pervect]

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".