Question about the metric tensor in Einstein's field equations.

In summary, the metric tensor has 10 parameters in total, with 3 representing each spatial dimension and 1 representing time. These parameters are derived from the symmetric and rank-2 nature of the metric tensor, and in a universe with T dimensions of time and S dimensions of space, there are (T+S)(T+S+1)/2 independent elements in the metric tensor. Each element represents a coefficient in Pythagoras' rule, with 3 representing distances in xyz directions and the remaining 6 representing other factors.
  • #1
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I was wonder if some can explain to me what exactly are the 10 parameters for the metric tensors. I know the reason for getting 10 parameters, 3^2=9 + 1, you get three for every spatial dimensions plus one for time. But why exactly three parameters for each spatial dimension? And what exactly are these three parameters for a spatial dimension? Can someone fill me in, I really what to know! :biggrin:
 
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  • #2
If you remember that the metric tensor is symmetric and rank-2, then in a spacetime with T dimensions of time and S dimensions of space, the number of independent elements of the metric (or "parameters" as you call them) is (T+S)(T+S+1)/2. For our particular universe, that works out to 4*5/2 = 10.
 
  • #3
Yes I know. But what exactly does each element represent? Like in a vector you have distance in the xyz directions. In a tensor I assume three of the elements are just distances in xyz but what about the other 6 elements? Sorry I'm trying to explain my question as best I can.
 
  • #4
The i-j element in the metric tensor is the coefficient of the dxidxj term in the general version of Pythagoras' rule. The tensor is symmetric because the multiplication is commutative, and dxidxj = dxjdxi.
 

1. What is the metric tensor in Einstein's field equations?

The metric tensor is a mathematical object used in Einstein's field equations to describe the curvature of spacetime. It is a matrix that contains information about the distances and angles in a given coordinate system.

2. How is the metric tensor related to gravity?

The metric tensor is used in Einstein's field equations to describe the relationship between the curvature of spacetime and the distribution of matter and energy. In other words, it is the mathematical representation of gravity in Einstein's theory of general relativity.

3. What does the metric tensor tell us about the shape of spacetime?

The metric tensor tells us about the curvature of spacetime. It describes how objects with mass and energy affect the shape of spacetime, which in turn determines how those objects move and interact with each other.

4. How is the metric tensor calculated?

The metric tensor is calculated using a mathematical formula that takes into account the coordinates of a given spacetime point and the distribution of matter and energy at that point. This calculation is integral to solving Einstein's field equations and understanding the effects of gravity.

5. Can the metric tensor change over time?

Yes, the metric tensor can change over time, particularly in situations where there is a significant change in the distribution of matter and energy. This change in the metric tensor can then affect the curvature of spacetime and the behavior of objects within it.

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