Assignment 2

Full Marks: 40

Due date: 20/02/2018

1. Find out the effective potential at the equatorial plane felt by atest particle moving in the spacetime

dτ 2 = (1−2Mr/Σ)dt2−Σ/∆ dr2−Σdθ2−(r2+a2+2Ma2r sin2 θ/Σ) sin2 θdφ2

+4aMr sin2 θ/Σ dtdφ,

where ∆ = r2 − 2Mr + a2, Σ = r2 + a2 cos2 θ, c = G = 1. Inaddition, find out the marginally (or innermost) stable circular orbitradius and marginally (or innermost) bound orbit radius in the abovespacetime. 4+5+5

2. If Ω = uφ/ut, prove that Ω =√M(r3/2 + a

√M)−1 for the above

spacetime. 4

3. Obtain the spherically symmetric solution of the Einstein’s equa-tion in vacuum in 2+1 dimension (i.e. 2 space-dimension). Interpretthe result. 3+2

4. Considering a metric in certain coordinates to the second ordergiven by

g00 = −1−R0l0mXlXm, g0i = g0i =

2

3R0limX

lXm,

gij = δij −1

3RiljmX

lXm,

where X i are the spatial coordinates of an event occurring at X0.What are the values of the Christoffel connections of this metric?Also obtain the components of Riemann tensor R1010, R0202, R1212,R2323, R2303, R2313 for the metric. 5+12