Understanding Einstein Field Equation & Metric Tensor

In summary: It's the same vector, with the same number of dimensions. The matching indices with the metric tensor tell you to sum over them.
  • #1
TimeRip496
254
5
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.
 
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  • #2
The metric tensor is a rank 2 symmetric covariant tensor. It doesn't a priori have anything to do with the Kronecker delta.
 
  • #3
Matterwave said:
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
 
  • #4
TimeRip496 said:
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.
 
  • #5
Matterwave said:
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.
 
  • #6
TimeRip496 said:
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.
 
  • #7
Matterwave said:
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.
 
  • #8
TimeRip496 said:
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:

$$s=\int_1^2 \sqrt{g_{ab}\frac{dx^a}{ds}\frac{dx^b}{ds}}ds$$
 
  • #9
TimeRip496 said:
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.
 
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  • #10
Matterwave said:
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:

$$s=\int_1^2 \sqrt{g_{ab}\frac{dx^a}{ds}\frac{dx^b}{ds}}ds$$
Matterwave said:
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 dxa and dxb, are they of the same vector but just with different no of dimensions?
 
  • #11
TimeRip496 said:
For your equation regarding the dxa and dxb, 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.
 

1. What is the Einstein Field Equation?

The Einstein Field Equation is a set of equations in the theory of general relativity that describes how the presence of matter and energy in the universe curves the fabric of space-time.

2. What is the significance of the Metric Tensor in the Einstein Field Equation?

The Metric Tensor is a mathematical object that represents the curvature of space-time in general relativity. It is used in the Einstein Field Equation to relate the distribution of matter and energy to the resulting curvature of space-time.

3. How does the Einstein Field Equation relate to the Theory of General Relativity?

The Einstein Field Equation is the cornerstone of the Theory of General Relativity, which is a theory of gravity that explains the behavior of large-scale objects in the universe. It describes how space-time is curved by matter and energy, and how this curvature affects the motion of objects in the universe.

4. What are the implications of understanding the Einstein Field Equation?

Understanding the Einstein Field Equation has many implications, including providing a deeper understanding of the nature of gravity and its effects on the universe. It also allows for the prediction and explanation of phenomena such as black holes, gravitational waves, and the expanding universe.

5. What are some real-world applications of the Einstein Field Equation?

The Einstein Field Equation has many practical applications, such as in the fields of astrophysics, cosmology, and space travel. It is also used in GPS systems, as the theory of general relativity accounts for the effects of gravity on the clocks used in GPS satellites.

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