Skip to main content
Physics

Return to Question

added 48 characters in body; edited tags
Source Link
Qmechanic ♦
Qmechanic ♦
  • 212.7k
  • 48
  • 589
  • 2.3k

I apologize for asking the duplicate question, but I still get confused after reading the related post In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are isometric, or just diffeomorphic?. In particular, for the statement

if universe is represented by manifold $M$ with matter fields $\psi$ and metric $g$, and $\phi: M \rightarrow M$ is a diffeomorphism, then sets $(M, g, \psi)$ and $(M, \phi^*g, \phi^* \psi)$ represents the same physical situation.

what does "same physical situation" really mean? Does it have some mathematical formulation? Is a diffeomorphism equivalent to a general coordinate transformation?

Also, in https://physics.stackexchange.com/a/596779, it said that we can actually prove the diffeomorphism invariance of the Hilbert action, can anyone gives a derivation?

I apologize for asking the duplicate question, but I still get confused after reading the related post. In particular, for the statement

if universe is represented by manifold $M$ with matter fields $\psi$ and metric $g$, and $\phi: M \rightarrow M$ is a diffeomorphism, then sets $(M, g, \psi)$ and $(M, \phi^*g, \phi^* \psi)$ represents the same physical situation.

what does "same physical situation" really mean? Does it have some mathematical formulation? Is a diffeomorphism equivalent to a general coordinate transformation?

Also, in https://physics.stackexchange.com/a/596779, it said that we can actually prove the diffeomorphism invariance of the Hilbert action, can anyone gives a derivation?

I apologize for asking the duplicate question, but I still get confused after reading the related post In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are isometric, or just diffeomorphic?. In particular, for the statement

if universe is represented by manifold $M$ with matter fields $\psi$ and metric $g$, and $\phi: M \rightarrow M$ is a diffeomorphism, then sets $(M, g, \psi)$ and $(M, \phi^*g, \phi^* \psi)$ represents the same physical situation.

what does "same physical situation" really mean? Does it have some mathematical formulation? Is a diffeomorphism equivalent to a general coordinate transformation?

Also, in https://physics.stackexchange.com/a/596779, it said that we can actually prove the diffeomorphism invariance of the Hilbert action, can anyone gives a derivation?

added 71 characters in body
Source Link
Hao
Hao
  • 169
  • 7

I apologize for asking the duplicate question, but I still get confused after reading the related post. In particular, for the statement

if universe is represented by manifold $M$ with matter fields $\psi$ and metric $g$, and $\phi: M \rightarrow M$ is a diffeomorphism, then sets $(M, g, \psi)$ and $(M, \phi^*g, \phi^* \psi)$ represents the same physical situation.

what does "same physical situation" really mean? Does it have some mathematical formulation? Is a diffeomorphism equivalent to a general coordinate transformation?

Also, in https://physics.stackexchange.com/a/596779, it said that we can actually prove the diffeomorphism invariance of the Hilbert action, can anyone gives a derivation?

I apologize for asking the duplicate question, but I still get confused after reading the related post. In particular, for the statement

if universe is represented by manifold $M$ with matter fields $\psi$ and metric $g$, and $\phi: M \rightarrow M$ is a diffeomorphism, then sets $(M, g, \psi)$ and $(M, \phi^*g, \phi^* \psi)$ represents the same physical situation.

what does "same physical situation" really mean? Does it have some mathematical formulation?

Also, in https://physics.stackexchange.com/a/596779, it said that we can actually prove the diffeomorphism invariance of the Hilbert action, can anyone gives a derivation?

I apologize for asking the duplicate question, but I still get confused after reading the related post. In particular, for the statement

if universe is represented by manifold $M$ with matter fields $\psi$ and metric $g$, and $\phi: M \rightarrow M$ is a diffeomorphism, then sets $(M, g, \psi)$ and $(M, \phi^*g, \phi^* \psi)$ represents the same physical situation.

what does "same physical situation" really mean? Does it have some mathematical formulation? Is a diffeomorphism equivalent to a general coordinate transformation?

Also, in https://physics.stackexchange.com/a/596779, it said that we can actually prove the diffeomorphism invariance of the Hilbert action, can anyone gives a derivation?

Source Link
Hao
Hao
  • 169
  • 7

Diffeomorphism invariance in GR

I apologize for asking the duplicate question, but I still get confused after reading the related post. In particular, for the statement

if universe is represented by manifold $M$ with matter fields $\psi$ and metric $g$, and $\phi: M \rightarrow M$ is a diffeomorphism, then sets $(M, g, \psi)$ and $(M, \phi^*g, \phi^* \psi)$ represents the same physical situation.

what does "same physical situation" really mean? Does it have some mathematical formulation?

Also, in https://physics.stackexchange.com/a/596779, it said that we can actually prove the diffeomorphism invariance of the Hilbert action, can anyone gives a derivation?