A DIAGONAL METRIC WORKSHEET
Consider the following general diagonal metric:
ds2 = –A(dx0)2 + B(dx1)2 + C(dx2)2 + D(dx3)2
where dx0, dx1, dx2, and dx3 are completely arbitrary coordinates and A, B, C, and D are arbitrary functions of
any or all of the coordinates. This worksheet (adapted from results listed in Rindler, Essential Relativity, 2/e,
a
Springer-Verlag, 1977) allows you to quickly calculate the components of Cno
and Rno /+ R a nao for any specific special case of such a metric. In this worksheet, I use the following shorthand notation:
2A
22 B
A0 / 0 , B12 / 1 2 , and so on.
2x
2x 2x
To use this worksheet, start by crossing out each tabulated term that is zero for the specific metric in question.
For the remaining terms, write the term’s value in the space above that term. For the Ricci tensor components,
you can then gather the terms in the space provided at the bottom. To adapt this worksheet to smaller dimensional spaces or spacetimes, treat the metric components corresponding to any nonexistent coordinates as if they
had the value 1 and the remaining metric components as being independent of the nonexistent coordinates.
CHRISTOFFEL SYMBOLS
0
C00
=
1
2A
A0
0
C100 = C01
=
0
C11
=
1
2A
B0
0
C22
=
1
2A
C0
0
C33
=
1
C11
=
1
2B
B1
1
1
C12
= C21
=
1
1
C01
= C10
=
1
C00
=
1
2B
1
2C
1
2D
A3
A1
1
C22
=- 21B C1
1
2C
C0
C122 = C221 =
1
2D
D0
C133 = C331 =
3
C11
=- 21D B3
C200 = C002 =
1
2A
1
2A
A2
1
2C
C1
C222 =
1
2C
1
2B
B2
D1
3
C22
=- 21D C3
RICCI TENSOR COMPONENTS (following three pages)
C113 = C311 =
C322 = C223 =
C2
C233 = C332 =
1
2A
A3
1
2B
B3
1
=0
other Cno
1
2C
2
=0
other Cno
2
C33
=- 21C D2
1
2D
C300 = C003 =
0
=0
other Cno
D0
1
C33
=- 21B D1
2
C11
=- 21C B2
A2
C033 = C330 =
3
C00
=
B0
A1
C022 = C220 =
2
C00
=
1
2B
1
2A
1
2D
D2
C333 =
1
2D
D3
3
=0
other Cno
C3
R00 = 0
+
1
2B
A11
+
1
2C
A22
+
1
2D
A33
B00
-
1
2C
C00
-
1
2D
D00
+
0
-
1
2B
+
0
+
1
4B 2
B 20
+
1
4C 2
C 20
+
1
4D 2
D 20
+
0
+
1
4AB
A0 B0
+
1
4AC
A0 C0
+
1
4AD
A0 D0
-
1
4BA
A1 A1
-
1
4B 2
A1 B1
+
1
4BC
A1 C1
+
1
4BD
A1 D1
-
1
4CA
A2 A2
+
1
4CB
A2 B2
-
1
4C 2
A2 C2
+
1
4CD
A2 D2
-
1
4DA
A3 A3
+
1
4DB
A3 B3
+
1
4DC
A3 C3
-
1
4D 2
A3 D3
B00
+
0
-
1
2C
B22
-
1
2D
B33
A11
+
0
-
1
2C
C11
-
1
2D
D11
0
+
1
4C 2
C 21
+
1
4D 2
D 21
+
1
4AC
B0 C0
+
1
4AD
B0 D0
+
1
4BC
B1 C1
+
1
4BD
B1 D1
R00 =
R11 =
1
2A
-
1
2A
+
1
4A 2
A 21
+
-
1
4A 2
B0 A0
-
+
1
4BA
B1 A1
+
-
1
4CA
B2 A2
+
1
4CB
B2 B2
+
1
4C 2
B2 C 2
-
1
4CD
B2 D2
-
1
4DA
B3 A3
+
1
4DB
B3 B3
-
1
4DC
B3 C3
+
1
4D 2
B3 D3
R11 =
1
4AB
B0 B0
0
R22 =
C00
-
1
2B
C11
+
0
-
1
2D
C33
A22
-
1
2B
B22
+
0
-
1
2D
D22
0
+
1
4D 2
D 22
C0 C0
+
1
4AD
C0 D0
C1 C1
-
1
4BD
C1 D1
+
1
4CD
C2 D2
C3 C3
+
1
4D 2
C3 D3
1
2A
-
1
2A
+
1
4A 2
A 22
+
1
4B 2
B 22
+
-
1
4A 2
C0 A0
+
1
4AB
C 0 B0
-
-
1
4BA
C1 A1
+
1
4B 2
C 1 B1
+
1
4BC
+
1
4CA
C2 A2
+
1
4CB
C2 B2
+
0
-
1
4DA
C3 A3
-
1
4DB
C3 B3
+
1
4DC
D00
-
1
2B
D11
-
1
2C
D22
+
0
A33
-
1
2B
B33
-
1
2C
C33
+
0
0
1
4AC
R22 =
R33 =
1
2A
-
1
2A
+
1
4A 2
A 23
+
1
4B 2
B 23
+
1
4C 2
C 23
+
-
1
4A 2
D0 A0
+
1
4AB
D0 B0
+
1
4AC
D0 C0
-
1
4AD
D0 D0
-
1
4BA
D1 A1
+
1
4B 2
D1 B1
-
1
4BC
D1 C1
+
1
4BD
D1 D1
-
1
4CA
D2 A2
-
1
4CB
D2 B2
+
1
4C 2
D2 C2
+
1
4CD
D2 D2
+
1
4DA
D3 A3
+
1
4DB
D3 B3
+
1
4DC
D3 C3
+
R33 =
0
R01 = 1
4AC
+
-
1
2D
A1 C0
+
1
4AD
1
2B
-
1
2D
+
1
4AD
-
1
2C
+
1
4AC
-
1
2D
+
1
4BD
-
1
2C
+
1
4BC
-
1
2B
+
1
4CB
1
2C
C01
D01
A1 D0
+
1
4C 2
C0 C1
+
1
4D 2
D0 D1
+
1
4BC
B0 C1
+
1
4BD
B0 D1
+
1
4B 2
B0 B2
+
1
4D 2
D0 D2
+
1
4CB
C 0 B2
+
1
4CD
C0 D2
+
1
4B 2
B0 B3
+
1
4C 2
C0 C3
+
1
4DB
D0 B3
+
1
4DC
D0 C3
+
1
4A 2
A1 A2
+
1
4D 2
D1 D2
+
1
4CA
C1 A2
+
1
4CD
C1 D2
+
1
4A 2
A1 A3
+
1
4C 2
C1 C3
+
1
4DA
D1 A3
+
1
4DC
+
1
4A 2
A2 A3
+
1
4B 2
B2 B3
+
1
4DA
D2 A3
+
1
4DB
D 2 B3
R01 =
R02 = +
1
4AB
B02
A2 B0
D02
A2 D0
R02 =
R03 = +
1
4AB
1
2B
B03
A3 B0
C03
A3 C0
R03 =
R12 = +
1
4BA
1
2A
A12
B2 A1
D12
B2 D1
R12 =
R13 = +
1
4BA
1
2A
A13
B3 A1
C13
B3 C 1
D1 C3
R13 =
R23 = +
R23 =
1
4CA
1
2A
A23
C3 A2
B23
C3 B2