- #1

Airsteve0

- 83

- 0

*"Einstein’s equations may be thought of as second-order differential equations for the*

metric tensor field gμν. There are ten independent equations (since both sides are symmetric

two-index tensors), which seems to be exactly right for the ten unknown functions of the

metric components. However, the Bianchi identity ∇μGμν = 0 represents four constraints on

the functions Rμν, so there are only six truly independent equations in (4.52). In fact this is

appropriate, since if a metric is a solution to Einstein’s equation in one coordinate system

xμ it should also be a solution in any other coordinate system xμ′ . This means that there are

four unphysical degrees of freedom in gμν (represented by the four functions xμ′ (xμ)), and

we should expect that Einstein’s equations only constrain the six coordinate-independent

degrees of freedom."

metric tensor field gμν. There are ten independent equations (since both sides are symmetric

two-index tensors), which seems to be exactly right for the ten unknown functions of the

metric components. However, the Bianchi identity ∇μGμν = 0 represents four constraints on

the functions Rμν, so there are only six truly independent equations in (4.52). In fact this is

appropriate, since if a metric is a solution to Einstein’s equation in one coordinate system

xμ it should also be a solution in any other coordinate system xμ′ . This means that there are

four unphysical degrees of freedom in gμν (represented by the four functions xμ′ (xμ)), and

we should expect that Einstein’s equations only constrain the six coordinate-independent

degrees of freedom."

[itex]R_{\mu\nu}[/itex]-[itex]\frac{1}{2}[/itex]R[itex]g_{\mu\nu}[/itex]=8[itex]\pi[/itex]G[itex]T_{\mu\nu}[/itex] (Eq. 4.52)

I am confused as to what the 6 independent equations represent. I'm getting lost on thinking that these equations represent 3 spatial coordinates and 1 time coordinate but then what do the other 2 components represent (obviously my understanding is flawed). Any clarification would be greatly appreciated, thanks.