Understanding Einstein Field Equation & Metric Tensor

[TimeRip496]

Hi guys. I am trying to understand einstein field equation and thus have started on learning tensor. For metric tensor, is it just comprised of two contra/covariant vectors tensor product among each other alone or does it requires an additional kronecker delta? I am confused about the idea behind kronecker delta and the metric tensor such what they are, in terms of physics.

 

[Matterwave]

The metric tensor is a rank 2 symmetric covariant tensor. It doesn't a priori have anything to do with the Kronecker delta.

 

[TimeRip496]

The metric tensor is a rank 2 symmetric covariant tensor. It doesn't a priori have anything to do with the Kronecker delta.

So it doesn't contain the kronecker delta? Cause I watch drphysics video regarding einstein field equation on YouTube and his metric tensor contains kronecker delta

 

[Matterwave]

So it doesn't contain the kronecker delta? Cause I watch drphysics video regarding einstein field equation on YouTube and his metric tensor contains kronecker delta

It doesn't have to contain a Kronecker delta, although it could.

 

[TimeRip496]

It doesn't have to contain a Kronecker delta, although it could.

So when does it contains? I am really confused as I don't really know about this kronecker delta.

 

[Matterwave]

So when does it contains? I am really confused as I don't really know about this kronecker delta.

The metric tensor tells you about distances on your manifold. More specifically, it is a rank 2 symmetric co variant tensor which allows you to find the inner product between any two vectors in the tangent space. What the metric tensor is for any specific manifold and matter distribution is the whole point of solving the Einstein Field Equations.

 

[TimeRip496]

The metric tensor tells you about distances on your manifold. More specifically, it is a rank 2 symmetric co variant tensor which allows you to find the inner product between any two vectors in the tangent space. What the metric tensor is for any specific manifold and matter distribution is the whole point of solving the Einstein Field Equations.

How does the inner product between two vecfors tells us about the distance on your manifold? This is the part which I am really confused about the metric tensor.

 

[Matterwave]

How does the inner product between two vecfors tells us about the distance on your manifold? This is the part which I am really confused about the metric tensor.

The inner product of a vector with itself basically just gives you the square of the magnitude of that vector. The infinitesimal length of tangent vectors are then added up over a curve to get the length of the curve:

$$s=\int_1^2 \sqrt{g_{ab}\frac{dx^a}{ds}\frac{dx^b}{ds}}ds$$

 

[Nugatory]

So it doesn't contain the kronecker delta? Cause I watch drphysics video regarding einstein field equation on YouTube and his metric tensor contains kronecker delta

You may want to try a more traditional written explanation of tensor calculus, instead of relying on YouTube videos. If you just want to understand what general relativity and the Einstein Field Equation is about, there is http://preposterousuniverse.com/grnotes/grtinypdf.pdf ... It cuts a fair number of mathematical corners, but will see you through to an understanding of how the concepts hang together.

 

[TimeRip496]

The inner product of a vector with itself basically just gives you the square of the magnitude of that vector. The infinitesimal length of tangent vectors are then added up over a curve to get the length of the curve:

$$s=\int_1^2 \sqrt{g_{ab}\frac{dx^a}{ds}\frac{dx^b}{ds}}ds$$

The inner product of a vector with itself basically just gives you the square of the magnitude of that vector. The infinitesimal length of tangent vectors are then added up over a curve to get the length of the curve:

$$s=\int_1^2 \sqrt{g_{ab}\frac{dx^a}{ds}\frac{dx^b}{ds}}ds$$

For your equation regarding the dx^a and dx^b, are they of the same vector but just with different no of dimensions?

 

[Matterwave]

For your equation regarding the dx^a and dx^b, are they of the same vector but just with different no of dimensions?

It's the same vector, with the same number of dimensions. The matching indices with the metric tensor tell you to sum over them.