Why should spacetime/manifold be smooth?

[mieral]

Why can't spacetime or manifold be crumpled like a piece of rug or paper.. why should it be smooth? Does it have to do with time or the geodesics discontinuous or time sporatic if it is not smooth?

 

[PeterDonis]

Why can't spacetime or manifold be crumpled like a piece of rug or paper.. why should it be smooth?

What exactly do you mean by "smooth" vs. "not smooth"? In GR, "smooth" means the manifold is differentiable, not that it can't be tightly curved. The reason for requiring differentiability is that without it we can't formulate differential equations, which are the kind of equations we use to construct models in physics (at least for cases where the things we're modeling are continuous). But you can have a differentiable manifold that can still be sharply curved; it can have "kinks" in it, as long as they're differentiable kinks. (And by the Einstein Field Equation, you would have to have an appropriate stress-energy tensor to cause the kinks.)

 

[mieral]

What exactly do you mean by "smooth" vs. "not smooth"? In GR, "smooth" means the manifold is differentiable, not that it can't be tightly curved. The reason for requiring differentiability is that without it we can't formulate differential equations, which are the kind of equations we use to construct models in physics (at least for cases where the things we're modeling are continuous). But you can have a differentiable manifold that can still be sharply curved; it can have "kinks" in it, as long as they're differentiable kinks. (And by the Einstein Field Equation, you would have to have an appropriate stress-energy tensor to cause the kinks.)

What is another synonym for "differentiable"? Is the following crumpled tissue differentiable?

What is an example of a manifold that is not differentiable?

 

[Drakkith]

What is another synonym for "differentiable"?

Do you understand what the mathematical concept of differentiation is?

 

[Nugatory]

What is another synonym for "differentiable"? Is the following crumpled tissue differentiable?

The wikipedia article on "differentiable manifolds" is an OK starting point... And the quick answer to the question about the tissue is that it is smooth.

 

[mieral]

I asked the following in a thread to Peterdonis.

"Please see graphics Are you saying it is possible to have sharp turns in spacetime rather than just smooth transition?"

Peter donis answered: "I'm not sure how you got that from what I said. To answer the question as you ask it, GR assumes that spacetime is always and everywhere a smooth manifold, so no."

Peterdonis answered "no" to question whether spacetime can take sharp turn.. But then in the crumpled tissue. it has many sharp turns!

Peterdonis.. in the image I asked you about.. how did you understand my question? See the following.. I was telling you the sharp turn was connected to the fabric.. it is continuous to the fabric.. so why did you say "no".. it is not smooth manifold? I got the impression you meant spacetime can only be smooth like surface of a bed and not a crumpled tissue.. but others say it is. So please emphasize how you understand the say following when the sharp turn is continuous to the fabric. Maybe you thought the peak in the pinkink thing was very sharp and hence not differentiable?

 

[PeterDonis]

how did you understand my question?

As being too vague to know for sure what you were describing. You didn't define what you meant by a "sharp turn", but you were contrasting it with being "smooth", so I took it to mean "not smooth". And GR does not allow spacetime to be "not smooth".

If "sharp turn" can be consistent with "smooth", then of course you can have "sharp turns" in a smooth manifold. But you shouldn't need me to tell you that.

The root of the problem here is that you are trying to use vague ordinary language instead of precise math. I strongly recommend learning the precise math. It will make discussions like this a lot easier.

 

[mieral]

As being too vague to know for sure what you were describing. You didn't define what you meant by a "sharp turn", but you were contrasting it with being "smooth", so I took it to mean "not smooth". And GR does not allow spacetime to be "not smooth".

If "sharp turn" can be consistent with "smooth", then of course you can have "sharp turns" in a smooth manifold. But you shouldn't need me to tell you that.

The root of the problem here is that you are trying to use vague ordinary language instead of precise math. I strongly recommend learning the precise math. It will make discussions like this a lot easier.

What kind of manifold in GR are not differentiable (not smooth).. do you mean the singularity of a black hole? What else?

 

[Nugatory]

I was telling you the sharp turn was connected to the fabric.. it is continuous to the fabric.. so why did you say "no".. it is not smooth manifold? I got the impression you meant spacetime can only be smooth like surface of a bed and not a crumpled tissue.. but others say it is.

Smoothness has nothing to do with bending or twisting or crumpling, and the surface of a bed is no more or less smooth than the surface of a crumpled piece of tissue.

Intuitively, if an arbitrarily small ant can crawl around on the surface of the tissue without ever having to jump, no matter how short the steps it is taking, then the surface of the tissue is smooth; and both a crumpled and an uncrumpled sheet of paper have that property. This is NOT a proper formal definition of smoothness, it's just a hint to give you a way of imagining what smoothness means. For the real thing, you'll have to look at the math, and the first few Google hits for "differentiable manifold" are pretty good.

 

[PeterDonis]

What kind of manifold in GR are not differentiable (not smooth)

None of them. GR only models smooth manifolds. As I've already said at least twice.

do you mean the singularity of a black hole?

No. The BH singularity is not part of the manifold. Neither are "singularities" in general.

 

[PAllen]

And just to point out, the crumpled tissue is actually perfectly flat and smooth. It's embedding is complicated, but the tissue's intrinsic geometry is unaffected by the embedding.

 

[mieral]

Does metric expansion in Big Bang occur because of time? Without time, the metric graph would just stop (which also means it is no longer smooth)?

 

[PAllen]

Does metric expansion in Big Bang occur because of time? Without time, the metric graph would just stop (which also means it is no longer smooth)?

In the geometric (manifold + metric) view of GR, metric expansion is a statement about the geometry of the Lorentzian manifold. Imagine a cone versus a sphere as 1 x 1 manifolds. For each, there exists a way of slicing them such that every slice is the same except for scale (circular slices). In the case of a cone, the circular slices grow forever, starting from the apex. For the sphere, the circular slices grow then shrink. There are analogous cases for FLRW manifolds. The metric expansion/contraction is a statement about the behavior of the spatial slices that are geometrically identical except for scale. Then, metric expansion means the scale grows if the singularity is placed in the past.

 

[mieral]

In the geometric (manifold + metric) view of GR, metric expansion is a statement about the geometry of the Lorentzian manifold. Imagine a cone versus a sphere as 1 x 1 manifolds. For each, there exists a way of slicing them such that every slice is the same except for scale (circular slices). In the case of a cone, the circular slices grow forever, starting from the apex. For the sphere, the circular slices grow then shrink. There are analogous cases for FLRW manifolds. The metric expansion/contraction is a statement about the behavior of the spatial slices that are geometrically identical except for scale. Then, metric expansion means the scale grows if the singularity is placed in the past.

Metric comes from the word meter.. which is usually connected to distance. If time is zero.. would there be any distance metric.. or does distance metric tensor needs time to even be defined?

 

[PAllen]

Metric comes from the word meter.. which is usually connected to distance. If time is zero.. would there be any distance metric.. or does distance metric tensor needs time to even be defined?

A Lorentzian metric has a signature which distinguishes timelike, lightlike, and spacelike directions. Other than that, I can not make any sense of your question.

 

[pervect]

Why can't spacetime or manifold be crumpled like a piece of rug or paper.. why should it be smooth? Does it have to do with time or the geodesics discontinuous or time sporatic if it is not smooth?

There are some proposals ("quantum foam") where the space-time manifold is not smooth, though I'm not sure if any of these proposals has been worked out well enough to make actual physical predictions. I believe they've been published, though I don't know the details (Wheeler was a proponent, IIRC).

Meanwhile, the approach where the space-time manifold is taken to be smooth works well for non-quantum gravity. It's a lot easier to deal with and DOES make actual physical predictions, which we have tested and which so far match our experimental results. So that's the theory we currently use for classical (i.e. non-quantum) gravity, one with smooth space-time manifold.

 

[PeterDonis]

Metric comes from the word meter.. which is usually connected to distance.

In ordinary language, yes. But physics does not use ordinary language. It uses technical language. "Metric" in physics is a technical term that has a precise meaning. Lorentzian metrics (which have timelike, spacelike, and null line elements) are included in that precise meaning.

 

[PeterDonis]

the metric graph would just stop (which also means it is no longer smooth)?

Spacetime manifolds are open sets, which means they will not "just stop".

 

[mieral]

Spacetime manifolds are open sets, which means they will not "just stop".

Let's take an example:

Today is March 7, 2017... the manifold above ends in March 2, 2017.. so time stopped there at March 2, 2017. What do you mean "Spacetime manifolds are open sets, which means they will not "just stop"? Thanks.

 

[PeterDonis]

Today is March 7, 2017... the manifold above ends in March 2, 2017..

This is not a valid solution because the manifold you drew is not an open set.

Also, you can't just declare by fiat that anything "just stops" at March 2, 2017 or any other point; you have to solve the Einstein Field Equation.

What do you mean "Spacetime manifolds are open sets

https://en.wikipedia.org/wiki/Open_set

 

[mieral]

A Lorentzian metric has a signature which distinguishes timelike, lightlike, and spacelike directions. Other than that, I can not make any sense of your question.

I know a Lorentzian metric is the distance inside the manifolds. You mentioned timelike, lightlike, and spacetime directions. I just want to know if it is possible to have timelike that has value zero yet non-zero value for lightlike, and spacetime. Can you point to some graphics about this. If impossible. Please tell me and emphasize "impossible" because you words are laws. Thanks.

 

[PAllen]

I know a Lorentzian metric is the distance inside the manifolds. You mentioned timelike, lightlike, and spacetime directions. I just want to know if it is possible to have timelike that has value zero yet non-zero value for lightlike, and spacetime. Can you point to some graphics about this. If impossible. Please tell me and emphasize "impossible" because you words are laws. Thanks.

I still cannot understand much of what you are asking. I will clarify about the distinction of different spacetime directions. Suppose you have an arbitrary curve in spacetime. At some point it has tangent vector. metrics can be written with either of two different conventions ( +,-,-,- or -,+,+,+ ). I will use the first convention. If the contraction of the tangent vector with metric is positive, it is timelike, and a possible way a body could move. If the contraction is zero, it is a possible way light could move. Otherwise, it is a possible simultaneity for some observer. In a lorentzian manifold, the metric at every point classifies possible tangent vectors into these three categories. All are possible at every point of the manifold.

 

[PeterDonis]

I just want to know if it is possible to have timelike that has value zero yet non-zero value for lightlike, and spacetime.

I think you are confused about what "timelike", "lightlike", and "spacelike" mean. The line element is an expression for $ds^2$ in terms of coordinate differentials. With the more common sign convention in GR, if $ds^2$ is negative, the interval is timelike; if $ds^2$ is zero, the interval is lightlike; if $ds^2$ is positive, the interval is spacelike. So "timelike that has value zero" and "non-zero value for lightlike" are contradictions in terms.

 

[PeterDonis]

The OP question has been answered. Thread closed.