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MeJennifer
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The pseudo metric of a Lorentzian space-time is an amalgamation of space and time.
So then what expands?
So then what expands?
As far as I understand it, it is definately not a metric.pervect said:the manifold of space-time is pseudo-Riemannian, but there's nothing "pseudo" about the metric. What separates a pseudo-Riemannian manifold from a Riemannian manifold is that the metric for the later must be positive definite. Because Lorentzian metrics have a -+++ signature they cannot be positive definite , due to the presence of the minus sign.
Yes I am familiar with it.pervect said:Write down a FRW metric and look at it, and tell us what you think expands. (I think you're capable of this, and it would be instructive for you, I think).
The components of the metric are a function of coordinates.MeJennifer said:The pseudo metric of a Lorentzian space-time is an amalgamation of space and time.
So then what expands?
MeJennifer said:The pseudo metric of a Lorentzian space-time is an amalgamation of space and time.
So then what expands?
Quite correct, so would that not imply that there must be one or more cross terms in the metric?pervect said:As far as the FRW metric goes, only the space terms get multipled by the expansion scale factor a(t).
Actually I am talking about the cross terms in the FWR metric not about the hypersurface.quantum123 said:Since the t curves are orthogonal to the hyper spatial surface, there is no cross terms.
MeJennifer said:As far as I understand it, it is definately not a metric.
For instance a metric must satisfy the triangle equality, clearly the space-time pseudo-metric does not.
MeJennifer said:The pseudo metric of a Lorentzian space-time is an amalgamation of space and time.
So then what expands?
Could you, in a nutshell, reason why you think one is a definitive example of space-time and the other definitively not?Chris Hillman said:You asked about the triangle inequality in Lorentzian metrics ("metric" in the algebraic sense of indefinite bilinear forms, not in the general topology sense of "metric toplogy"!):
MeJennifer said:So the components are pure time and space and do not contain any cross terms?
MeJennifer said:Could you, in a nutshell, reason why you think one is a definitive example of space-time and the other definitively not?
Chris Hillman said:Note that not all Lorentzian spacetimes are "diagonalizable" in this sense. For example, exact solutions which model a circularly polarized gravitational wave propagating far from any massive objects are not "diagonalizable". QUOTE]
So you mean in such a region of spacetime, there can't be free falling observers?
hellfire said:I may be wrong but to me it seams more a matter of interpretation than only a matter of a mere mathematical derivation of the FRW metric, because in an FRW metric you can always perform a coordinate change to conformal coordinates that give you expanding space and expanding time (conformal time). I understand that with expanding you mean some coordinate that is multiplied by the scale factor. However, this conformal time does not correspond to our proper physical time that we measure locally with clocks.
Well to me "general topologyl" and Riemannian manifolds are not identical. Riemannian manifolds are simply a subset. I had, perhaps falsely, the impression that you exclude the possibility that space-time can be modeled topologically. That was why I asked the particular question. But if you did not then I apologize.Chris Hillman said:I said no such thing!
I did not misread and I really meant definitively.Chris Hillman said:(If I hadn't seen previous posts by you, I'd think you were trolling, so in future, for all our sakes, please try harder not to misread standard technical terms as informal terms in natural language).
MeJennifer said:Well to me "general topological" and Riemannian manifolds are not identical.
MeJennifer said:Riemannian manifolds are simply a subset.
I had the impression that you exclude the possibility that space-time can be modeled topologically.
I did not misread and I really meant definitively.
But if you did not then I apologize.
Metric expansion in Lorentzian space-time refers to the phenomenon where the distances between objects in the universe increase over time. This is due to the expansion of the universe itself, as described by the theory of general relativity.
Metric expansion has several effects on the universe. It causes the redshift of light from distant objects, making them appear to be moving away from us. It also affects the overall geometry of the universe, leading to the formation of structures such as galaxies and clusters of galaxies.
Dark energy is a hypothetical form of energy that is thought to be responsible for the acceleration of metric expansion. It is believed to make up a significant portion of the total energy density of the universe, but its exact nature is still not fully understood.
Yes, metric expansion can be observed through various observations, such as the redshift of light from distant objects and the distribution of galaxies in the universe. These observations provide evidence for the expansion of the universe and the role of metric expansion in shaping its structure.
No, metric expansion is not constant. It has been observed to accelerate over time, which is believed to be due to the influence of dark energy. However, the rate of acceleration is not constant and may change in the future, leading to different scenarios for the ultimate fate of the universe.