A Flow as a fluid-orthogonal foliation

[Apashanka]

If considering the 4-D manifold($x^1,x^2,x^3,t$) in which the sub-manifold dimensionality is 3($x^1,x^2,x^3$)(hypersurface)
We are writting the metric of this space as $ds^2=g(t)dt^2+f(t)\gamma_{ij}{dx^idx^j}\delta_{ij}$ where the term $\gamma_{ij}dx^idx^j\delta_{ij}$ is the metric of the hypersurface assuming that $\frac{∂}{∂t}=\hat t$ is orthogonal to the hypersurface .
A very elementary rough diagram is this

Now from the four velocity $V^\mu=\gamma${c,$v_{spatial}$}
If a particle is at rest in one frame then it's $V^\mu$ points along $\hat t$ (perpendicular to the hypersurface) and if it is having some spatial velocity in another frame then the velocity four vector $V^\mu$ is now tilted at an angle to the hypersurface(e.g roughly we can say there are two components ,along $\hat t$ and $V_{hypersurface})$
Till now it's ok but these lines from a paper I am unable to understand and connects ,to the above

From ''Accordingly .....to $b_{\mu \nu}$'' here $n$ is along $\hat t$ and $u$ is the four velocity.
Can anyone help me in sort out this

 

[PeterDonis]

a paper

What paper? Please give a reference.

 

[Apashanka]

What paper? Please give a reference.

https://www.google.com/url?sa=t&sou...FjAMegQIBhAB&usg=AOvVaw3_pDp4cpP_IVxhDOjPw3P-

 

[PeterDonis]

Moderator's note: Thread level changed to "A" based on the reference given.

 

[Apashanka]

If considering the 4-D manifold($x^1,x^2,x^3,t$) in which the sub-manifold dimensionality is 3($x^1,x^2,x^3$)(hypersurface)
We are writting the metric of this space as $ds^2=g(t)dt^2+f(t)\gamma_{ij}{dx^idx^j}\delta_{ij}$ where the term $\gamma_{ij}dx^idx^j\delta_{ij}$ is the metric of the hypersurface assuming that $\frac{∂}{∂t}=\hat t$ is orthogonal to the hypersurface .
A very elementary rough diagram is this View attachment 239632
Now from the four velocity $V^\mu=\gamma${c,$v_{spatial}$}
If a particle is at rest in one frame then it's $V^\mu$ points along $\hat t$ (perpendicular to the hypersurface) and if it is having some spatial velocity in another frame then the velocity four vector $V^\mu$ is now tilted at an angle to the hypersurface(e.g roughly we can say there are two components ,along $\hat t$ and $V_{hypersurface})$
Till now it's ok but these lines from a paper I am unable to understand and connects ,to the aboveView attachment 239633
From ''Accordingly .....to $b_{\mu \nu}$'' here $n$ is along $\hat t$ and $u$ is the four velocity.
Can anyone help me in sort out this

Can anyone please help me...

 

[martinbn]

What exactly is the question? In that sentence there are no computations or any arguments/reasoning to explain. They only introduce the standard notations and names for the projections.

 

[Apashanka]

What exactly is the question? In that sentence there are no computations or any arguments/reasoning to explain. They only introduce the standard notations and names for the projections.

I didnt understand the projections they're taking about??

 

[PeterDonis]

I didnt understand the projections they're taking about??

If you don't understand the general concept of projections, then the paper you referenced is probably too advanced for you. And trying to explain that concept from scratch is probably too much for a PF thread.

What textbooks on differential geometry have you studied? Projections are a basic concept of differential geometry. You really need to have a good understanding of differential geometry before trying to work through a paper like the one you linked to.

 

[PeterDonis]

I didnt understand the projections they're taking about??

The basic idea is that, if you have a timelike vector at a given event, then the metric of a local spacelike hypersurface orthogonal to that timelike vector at that event is given by the formulas in what you quoted. The formulas are given for two different choices of timelike vector: $n$, the vector that points along the timelike lines of the chosen foliation, and $u$, the vector that points along the fluid flow lines.

Going into much more detail than that is probably, as I said before, beyond the scope of a PF thread.

 

[Apashanka]

Sir one question, space like vectors can't be along $\hat t$ ,isn't it??they are outside the light cones

 

[PeterDonis]

space like vectors can't be along $^\hat{t}$ ,isn't it??

Not if $\hat{t}$ is timelike, no.

 

[PeterDonis]

Thread closed as the subject matter is "A" level and the OP does not appear to have the required background.

@Apashanka, based on your repeated threads on subjects related to this one, it really appears that you need to spend some time developing your understanding of differential geometry and how it is used in General Relativity. Here at PF we can help with specific questions, but we are not equipped to give a complete graduate level course in differential geometry and GR, and it's difficult to answer questions that require that background knowledge if the person asking them does not have it.