# This script calculates the Einstein tensor for a static spherically symmetric
# metric.
#
# Cf. "A first course in general relativity," Bernard F. Schutz, p. 255.
#
# This is the line element for the metric (Equation 10.7)
#
# 2 2 Phi 2 2 Lambda 2 2 2
# ds = -e dt + e dr + r d Omega
#
# where
#
# 2 2 2 2 2 2
# r d Omega = r (d theta + sin theta d phi )
#
# Note: Phi and Lambda are both functions of r.
gdd = ((-exp(2 Phi(r)), 0, 0, 0),
( 0, exp(2 Lambda(r)), 0, 0),
( 0, 0, r^2, 0),
( 0, 0, 0, r^2 sin(theta)^2))
# Note: "dd" stands for two "down" indices, "uu" stands for two "up" indices.
# X is our coordinate system. We need this for computing gradients.
X = (t,r,theta,phi)
# Step 1: Calculate guu.
guu = inv(gdd)
# Step 2: Calculate the connection coefficients. Cf. Gravitation, p. 210.
#
# Gamma = 1/2 (g + g - g )
# abc ab,c ac,b bc,a
#
# Note: The comma means gradient which increases the rank of gdd by 1.
gddd = d(gdd,X)
# Note: We transpose indices so they match up with Gamma, i.e., we put them in
# alphabetical order.
GAMDDD = 1/2 (gddd + # indices are already in correct order
transpose(gddd,2,3) - # transpose c and b
transpose(transpose(gddd,2,3),1,2)) # transpose c and a, then b and a
# Raise first index.
#
# a au
# Gamma = g Gamma
# bc ubc
#
# Note: Sum over index u means contraction.
GAMUDD = contract(outer(guu,GAMDDD),2,3)
# Step 3. Calculate the Riemann tensor. Cf. Gravitation, p. 219.
#
# a is alpha
# b is beta
# c is gamma
# d is delta
# u is mu
#
# a a a a u a u
# R = Gamma - Gamma + Gamma Gamma - Gamma Gamma
# bcd bd,c bc,d uc bd ud bc
#
# Do the gradient once and save in a temporary variable.
T1 = d(GAMUDD,X)
# The Gamma Gamma product is a rank 6 tensor with dim 4 per rank.
# That works out to 4 to the 6th or 4,096 elements.
# Of course, we'll do the outer product and contract over u just once and save
# the result in a second temporary variable.
T2 = contract(outer(GAMUDD,GAMUDD),2,4)
# Now put it all together. Do the transpositions so the indices get matched up
# with R on the left, i.e., put them in alphabetical order.
RUDDD = transpose(T1,3,4) - # transpose d and c
T1 + # already in correct order
transpose(T2,2,3) - # transpose c and b
transpose(transpose(T2,2,3),3,4) # transpose d and b, then d and c
# Step 4: Calculate the Ricci tensor. Cf. Gravitation, p. 343.
#
# a
# R = R
# uv uav
#
# Contract over "a" (1st and 3rd indices).
RDD = contract(RUDDD,1,3)
# Step 5: Calculate the Ricci scalar. Cf. Gravitation, p. 343.
#
# uv
# R = g R
# vu ...the book has uv, does it give the same result?
# Yes because the metric tensor is symmetric so it's ok to
# transpose.
R = contract(contract(outer(guu,RDD),2,3),1,2)
# Step 6: Finally, calculate the Einstein tensor. Cf. Gravitation, p. 343.
#
# G = R - 1/2 g R
# uv uv uv
GDD = RDD - 1/2 gdd R
# Next we compare this result with Schutz' book. Schutz p. 255 gives the
# following Einstein tensor components (all other components are zero):
#
# 1 d
# G = ---- exp(2 Phi) ---- [r (1 - exp(-2 Lambda))]
# tt 2 dr
# r
#
# 1 2
# G = - ---- exp(2 Lambda) (1 - exp(-2 Lambda)) + --- Phi'
# rr 2 r
# r
#
# 2 2
# G = r exp(-2 Lambda) [Phi'' + (Phi') + Phi'/r
# theta theta
#
# - Phi' Lambda' - Lamda'/r]
#
# 2
# G = sin theta G
# phi phi theta theta
Gtt = 1/r^2 exp(2 Phi(r)) d(r (1 - exp(-2 Lambda(r))),r)
Grr = -1/r^2 exp(2 Lambda(r)) (1 - exp(-2 Lambda(r))) + 2/r d(Phi(r),r)
Gthetatheta = r^2 exp(-2 Lambda(r)) (
d(d(Phi(r),r),r) +
d(Phi(r),r)^2 +
d(Phi(r),r) / r -
d(Phi(r),r) d(Lambda(r),r) -
d(Lambda(r),r) / r)
Gphiphi = sin(theta)^2 Gthetatheta
# Put together the expected tensor:
T = ((Gtt, 0, 0, 0),
( 0, Grr, 0, 0),
( 0, 0, Gthetatheta, 0),
( 0, 0, 0, Gphiphi))
# Check that GDD is correct.
check(GDD = T)
# Display the non-zero components of GDD.
Gtt
Grr
Gthetatheta
Gphiphi