Hi all,
I am trying to find the Christoffel connections of this metric:
ds2= -(1+2∅)dt2 +(1-2∅)[dx2+dy2+dz2]
where ∅ is a general function of x,y,z,t.
I tried to solve this through the least action principle, but some of my results(t-related terms) were different from the answer with a minus...
Sorry for that I did not write down the other non-zero terms of Riemann curvature tensor which can be deduced by symmetry and anti-symmetry properties.
However, I still have a contradiction that
Rt _ztz-b''(z)-[b'(z)]2
but
Rz_tzt=[b''(z)+[b'(z)]2]e2b(z)
Did you also get the same result?
Need to find the Ricci scalar curvature of this metric:
ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor:
<The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z...
Homework Statement
Need to find the Ricci scalar curvature of this metric:
ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2
Homework Equations
The Attempt at a Solution
I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor:
<The Christoffel...
I got some trouble from this question:
For a given metric: ds2 =t-2(dx2-dt2), derive the energy-momentum tensor which satisfies the Einstein equation: Rαβ- 1/2Rgαβ=8\piGTαβ.
I got the Ricci scalar R=2, but Tαβ=0 for all α,β. Does this means a curved spacetime without any source(energy-momentum...