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mieral
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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?
mieral said:Why can't spacetime or manifold be crumpled like a piece of rug or paper.. why should it be smooth?
PeterDonis said: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 said:What is another synonym for "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 said:What is another synonym for "differentiable"? Is the following crumpled tissue differentiable?
mieral said:how did you understand my question?
PeterDonis said: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.
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.mieral said: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.
mieral said:What kind of manifold in GR are not differentiable (not smooth)
mieral said:do you mean the singularity of a black hole?
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 said: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 said: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.
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.mieral said: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?
mieral said: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?
mieral said:Metric comes from the word meter.. which is usually connected to distance.
mieral said:the metric graph would just stop (which also means it is no longer smooth)?
PeterDonis said:Spacetime manifolds are open sets, which means they will not "just stop".
mieral said:Today is March 7, 2017... the manifold above ends in March 2, 2017..
mieral said:What do you mean "Spacetime manifolds are open sets
PAllen said: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 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.mieral said: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.
mieral said:I just want to know if it is possible to have timelike that has value zero yet non-zero value for lightlike, and spacetime.
The smoothness of spacetime/manifold is important in physics because it allows us to understand and describe the behavior of matter and energy in our universe. A smooth spacetime/manifold allows for the smooth propagation of light and other particles, making it possible to accurately predict and measure the effects of gravity and other physical phenomena.
The theory of relativity, both special and general, relies on the concept of a smooth spacetime/manifold. In special relativity, the smoothness of spacetime is necessary for the constancy of the speed of light, a fundamental principle of the theory. In general relativity, the smoothness of spacetime is essential for understanding the curvature of space caused by the presence of mass and energy.
While spacetime/manifold can be considered smooth at macroscopic scales, at very small scales, such as the Planck length, it is believed to become "foamy" or "quantized". This is due to the uncertainty principle and the effects of quantum mechanics, which introduce a fundamental graininess to the fabric of spacetime.
If spacetime/manifold were not smooth, it would significantly impact our understanding of the laws of physics and our ability to make accurate predictions about the behavior of matter and energy. The smoothness of spacetime is essential for many fundamental physical principles, such as the conservation of energy and momentum, and the behavior of particles and waves.
Yes, there is significant evidence for the smoothness of spacetime/manifold. This includes the successful predictions of the theory of relativity, the observation of the bending of light by massive objects, and the precise measurements of the cosmic microwave background radiation. Additionally, experiments such as the Michelson-Morley experiment and the LIGO gravitational wave detector also support the concept of a smooth spacetime/manifold.