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

[Entropy]

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!

 

[Janitor]

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.

 

[Entropy]

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.

 

[selfAdjoint]

The i-j element in the metric tensor is the coefficient of the dx_idx_j term in the general version of Pythagoras' rule. The tensor is symmetric because the multiplication is commutative, and dx_idx_j = dx_jdx_i.