How many numbers to describe a point in space in GR?

[edpell]

How many numbers does it take to describe a point in space in GR?

 

[hamster143]

Just the location? Four.

 

[edpell]

Just the location? Four.

I guess I set myself up for that one. Not sure how to word it. With all the stress and strain and tension and whatever I have missed what does it take to describe a point in space? Maybe I should say describe all the things that go into Einstein's equation

R_{ab} - {\textstyle 1 \over 2}R\,g_{ab} = \kappa T_{ab}.\,

 

[diazona]

I'm really not sure what you're getting at. It takes only 4 coordinates (numbers) to uniquely identify any point in space and time; that's why we call it a 4-dimensional spacetime. But there is a lot of extra information that goes into Einstein's equation that is not built into the 4 coordinates of a point - like the energy density at the point, the curvature at the point, etc. None of that changes the fact that 4 coordinates are enough to uniquely identify the point.

Can you explain your question in some more detail, perhaps?

 

[edpell]

there is a lot of extra information that goes into Einstein's equation that is not built into the 4 coordinates of a point - like the energy density at the point, the curvature at the point, etc.

I am asking what are all the "extra information that goes into Einstein's equation"?

you have:
energy density (one number? a four vector?)
curvature (one number? a four vector?)
others?

Thanks.

 

[Spinnor]

How many numbers does it take to describe a point in space in GR?

Check out the old thread:

https://www.physicsforums.com/showthread.php?t=346393

Look at post # 14 by Phark:

"...I found something on it. The wiki claims that a pseudo-Riemann manifold can be embedded in an n(n+1)/2 pseudo-Euclidian space. This would be 10 for spacetime. ..."

Also if the metric g_ij is a symetric 4X4 tensor then at a point it can be approximated by ten numbers?

 

[Nabeshin]

Indeed, one can completely determine the metric at any point in spacetime with 10 numbers. If one allows for coordinate transformations, a clever person could get this down to only six numbers.

 

[edpell]

Indeed, one can completely determine the metric at any point in spacetime with 10 numbers. If one allows for coordinate transformations, a clever person could get this down to only six numbers.

Not being smart I will stick with 10.

So each point can be described by four numbers for the coordinate and 10 numbers for one symmetric 4x4 tensor a total of 14 numbers.

 

[hamster143]

Not being smart I will stick with 10.

So each point can be described by four numbers for the coordinate and 10 numbers for one symmetric 4x4 tensor a total of 14 numbers.

A well posed question is half the answer.

The question "where is it" can be answered with four numbers, x, y, z, t, presupposing the coordinate system.

The question "what's the relationship between distances and the coordinate chart in the vicinity of that point" can be answered with ten numbers, ten components of the metric tensor, again presupposing the coordinate system.

The question "what's the geometry and the curvature in the vicinity of that point" can be answered with twenty additional numbers, due to the fact that the most general Riemann tensor has twenty independent components.

The question "what's the physics over there" requires four to infinity numbers, four in the simplest case of non-interacting dust, physically interesting cases will, at the very least, require pressure, viscosity strains and temperature, for the total of 11 (10 independent components of the stress-energy tensor + temperature). If you have multiple fields in the same region of space, you need the complete description of all internal degrees of freedom for each.

 

[edpell]

I am beginning to see why numerical GR (computer simulations) is still just in the starting phase.

 

[Spinnor]

A well posed question is half the answer.

The question "where is it" can be answered with four numbers, x, y, z, t, presupposing the coordinate system.

The question "what's the relationship between distances and the coordinate chart in the vicinity of that point" can be answered with ten numbers, ten components of the metric tensor, again presupposing the coordinate system.

The question "what's the geometry and the curvature in the vicinity of that point" can be answered with twenty additional numbers, due to the fact that the most general Riemann tensor has twenty independent components.

The question "what's the physics over there" requires four to infinity numbers, four in the simplest case of non-interacting dust, physically interesting cases will, at the very least, require pressure, viscosity strains and temperature, for the total of 11 (10 independent components of the stress-energy tensor + temperature). If you have multiple fields in the same region of space, you need the complete description of all internal degrees of freedom for each.

What a thought provoking answer, thank you!