Numerical Relativity - PHY 6938
Solutions to HW 6
1. This problem is actually quite lengthy. Here we give only the outline of a straightforward
brute force attack.
Start with
Rab = c c a c + c d c d
ab
cb
ab cd
db ca
rewrite in terms of abc :
Rab = c (g c
Numerical Relativity - PHY 6938
Solutions to HW 5
l
1. Recall that two covariant derivative operators dier by a tensor Cik , e.g.
l
Di wk = Di wk Cik wl .
Read Walds book if you do not believe this.
Lets start with
Rijk l wl = Di Dj wk Dj Di wk
m
l
m
l
Numerical Relativity - PHY 6938
Solutions to HW 4
f g h
d e
1. Lets start with (3)Rabcd wd = Da Db wc Db Da wc and note Da Db wc = a b c f (g h d we ).
f g
f g
d
f g
d
f g
Now we use a b f g = a b f (g + ng nd ) = a b (nd f ng + ng f nd ) = a b (nd f ng +
Numerical Relativity - PHY 6938
HW 3
Hand in this homework.
READ: Chap 2.1-2.4
PROBLEMS:
1. In units where G = c = 1 we can express all physical quantities in units of meters.
a) Express the following in meters: 1s, 1kg, 1m/s, 1m/s2
b) In units of G = c =
Numerical Relativity - PHY 6938
HW 6
Hand in this homework.
READ: Chap 4., Appendix B.
PROBLEMS:
1. Show that the Ricci tensor can be written as
1
Rab = g cd c d gab + (a b) + g cd g ef e gca f gdb ace bdf ,
2
(1)
in any coordinate system. Here
a b
:= a b
Numerical Relativity - PHY 6938
Solutions to HW 9
1. Recall dE = T dS pdV + dN and E = T S pV + N
a)
d = dE/V E/V 2 dV = (dE dV )/V
dS = N ds + sdN
dN = V dn + ndV
Thus d = [T (N ds+sdN )pdV +(V dn+ndV )dV ]/V = [T (N ds+s(V dn+ndV )pdV +
(V dn + ndV ) dV
Numerical Relativity - PHY 6938
HW 10
Hand in this homework.
READ: Chap 7, 9.
PROBLEMS:
1. Consider an advection equation of the form
t u + vx u = 0
In homework 8 we have seen that the discretization
un+1 = un
m
m
vt n
(u
un )
m1
2x m+1
1
is unstable. R
Numerical Relativity - PHY 6938
HW 9
Hand in this homework.
READ: Chap 7.
PROBLEMS:
1. The rst law of thermodynamics is usually written as
dE = T dS pdV + dN
Here E is the total relativistic energy.
a) Dene := E/V , n := N/V , s := S/N = S/(V n). Calculat
Numerical Relativity - PHY 6938
Solutions to HW 10
1.
a) The Lax method is second order accurate in space. If we use Taylor expansions of the
RHS and trade spatial for time derivs, we nd
1 2
1 x2
un+1 = un + x un x2 +O(x3 )vtx un +O(tx2 ) = un + 2 2 un t2
Numerical Relativity - PHY 6938
Solutions to HW 8
1. a) In this problem we are trying to solve the PDE t u + x u = 0. This is a single mode
with speed 1 moving to the right. I.e. on the left boundary (x = 0) we need to specify a
boundary condition to say
Numerical Relativity - PHY 6938
HW 1
Do not hand in this homework.
READ: Chap 1.1-1.9
PROBLEMS:
Do the following problems from the problem book: 8.16, 10.3, 10.10, 10,11
1
Numerical Relativity - PHY 6938
HW 7
Hand in this homework.
READ: Chap 5.
PROBLEMS:
1. Its time to do some numerics. Here we use Python
(see e.g. https:/docs.python.org/3/tutorial/index.html) because its so simple that
for what we do, you do not even need
Numerical Relativity - PHY 6938
HW 2
Do not hand in this homework.
READ: Chap 1.10-1.17
PROBLEMS:
Do the following problems from the problem book: 5.3, 5.18, 5.19, 5.21, 14.3
1
Numerical Relativity - PHY 6938
HW 4
Hand in this homework.
READ: Chap 2.5-2.9
PROBLEMS:
1. Derive the Gau-Codazzi and Codazzi-Mainardi equations.
2. Derive the relation
c
c d
a b ne nf Rcedf = n Kab + Kac Kb + (Da Db )/
from
(
a
b
b
a )nc
= Rabcd nd .
3.
Numerical Relativity - PHY 6938
Solutions to HW 7
1.
b) In this problem we are trying to solve the PDE t u + x u = 0 on a grid with grid points
xi = dx i, where dx = 0.1 and i 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
c) When we run the program it prints the valu
Numerical Relativity - PHY 6938
HW 5
Hand in this homework.
READ: Chap 3., Appendix A.
PROBLEMS:
1. Let ij = 4 ij . Express the 3D Ricci scalar R (computed from ij ) in terms of R
ij
(computed from ) and .
(It would be useful if you also gave expressions
Numerical Relativity - PHY 6938
HW 8
Hand in this homework.
READ: Chap 5. & 6.
PROBLEMS:
1. Last time you studied and ran the python program advection1.py.
a) advection1.py contains 3 functions to take spatial derivatives, namely Dm, Dp, D0. Try
all 3 in