References
17
A y 1 ...y n :=
B y 1 ...y n
,
(1.87)
A y 1 ...y n+1 :=
B y 1 ...y n ;y n+1
− A y 1 ...y n ;y n+1 ,
(1.88)
A y 1 ...y n+2 :=
B y 1 ...y n ;y n+1 y n+2
− A y 1 ...y n y 0
σ
y 0 y n+1 y n+2
−A y 1 ...y n ;y n+1 y n+2 − 2A y 1 ...y n (y n+1 ;y n+2 ) .
(1.89)
With the help of (1.86) we are able to iteratively expand any bitensor to any
order, provided the coincidence limits entering the expansion coefficients can be
calculated. The expansion for bitensors with mixed index structure can be obtained
from transporting the indices in (1.86) by means of the parallel propagator. We do
not give the explicit form of the expansions here (they can be found in [19]) but only
state the final result in the form of the expanded version of (1.85) up to the second
order:
D 2
d ˜
t 2 η
y 1 = ˜
a
y 1 − a
y 1 − η
y 4 R
y 1 y 2 y 3 y 4
u
y 2 u
y 3 + 2u
y 3
Dη y 2
d ˜
t
+η
y 4 η
y 5
u
y 2 u
y 3
1
2
∇ y 2 R
y 1 y 4 y 5 y 3 −
1
3
∇ y 4 R
y 1 y 2 y 3 y 5
+
1
3
R
y 1 y 4 y 5 y 2
a
y 2 +
1
2
˜
a
y 2
+ O(σ
3 ).
(1.90)
It becomes clear by specialization to geodesic world lines that the use of systematic
expansions of the exact deviation equation in powers of the derivatives of the world
function allows for a recovery of the well-known geodesic deviation equation, as
given in (1.39), which is represented by the third term in first line on the rhs of
(1.90).
Finally we note that the explicit form of the expanded generalized deviation
equation up to the third order can be found in [19]. There is even a generalization of
the deviation equation available for Riemann-Cartan geometries [20].
References
1. E. Fermi, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. Rend. 31, 2151101 (1922)
2. J.L. Synge, Relativity: The General Theory (North-Holland, Amsterdam, 1960)
3. J. Ehlers, Akad. Wiss. Mainz Abh. Math.-Naturw. Kl. 11, 763 (1961)
4. J. Ehlers, Gen. Rel. Grav. 25, 1225 (1993)
5. P. Jordan, J. Ehlers, R.K. Sachs, Akad. Wiss. Mainz, Abh. Math.-Naturw. Kl. 1, 1 (1961)
6. P. Jordan, J. Ehlers, R.K. Sachs, Gen. Rel. Grav. 45, 2691 (2013)
7. F.A.E. Pirani, Brandeis Lectures (Prentice Hall, New Jersey, 1965), p. 249
8. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, Oxford,
1983)
9. B.S. DeWitt, R.W. Brehme, Ann. Phys. 9, 220 (1960)
10. E. Poisson, A. Pound, I. Vega, Living Rev. Relativ. 14(7) (2011)
11. W.G. Dixon, Nuovo Cimento 34, 317 (1964)
17
A y 1 ...y n :=
B y 1 ...y n
,
(1.87)
A y 1 ...y n+1 :=
B y 1 ...y n ;y n+1
− A y 1 ...y n ;y n+1 ,
(1.88)
A y 1 ...y n+2 :=
B y 1 ...y n ;y n+1 y n+2
− A y 1 ...y n y 0
σ
y 0 y n+1 y n+2
−A y 1 ...y n ;y n+1 y n+2 − 2A y 1 ...y n (y n+1 ;y n+2 ) .
(1.89)
With the help of (1.86) we are able to iteratively expand any bitensor to any
order, provided the coincidence limits entering the expansion coefficients can be
calculated. The expansion for bitensors with mixed index structure can be obtained
from transporting the indices in (1.86) by means of the parallel propagator. We do
not give the explicit form of the expansions here (they can be found in [19]) but only
state the final result in the form of the expanded version of (1.85) up to the second
order:
D 2
d ˜
t 2 η
y 1 = ˜
a
y 1 − a
y 1 − η
y 4 R
y 1 y 2 y 3 y 4
u
y 2 u
y 3 + 2u
y 3
Dη y 2
d ˜
t
+η
y 4 η
y 5
u
y 2 u
y 3
1
2
∇ y 2 R
y 1 y 4 y 5 y 3 −
1
3
∇ y 4 R
y 1 y 2 y 3 y 5
+
1
3
R
y 1 y 4 y 5 y 2
a
y 2 +
1
2
˜
a
y 2
+ O(σ
3 ).
(1.90)
It becomes clear by specialization to geodesic world lines that the use of systematic
expansions of the exact deviation equation in powers of the derivatives of the world
function allows for a recovery of the well-known geodesic deviation equation, as
given in (1.39), which is represented by the third term in first line on the rhs of
(1.90).
Finally we note that the explicit form of the expanded generalized deviation
equation up to the third order can be found in [19]. There is even a generalization of
the deviation equation available for Riemann-Cartan geometries [20].
References
1. E. Fermi, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. Rend. 31, 2151101 (1922)
2. J.L. Synge, Relativity: The General Theory (North-Holland, Amsterdam, 1960)
3. J. Ehlers, Akad. Wiss. Mainz Abh. Math.-Naturw. Kl. 11, 763 (1961)
4. J. Ehlers, Gen. Rel. Grav. 25, 1225 (1993)
5. P. Jordan, J. Ehlers, R.K. Sachs, Akad. Wiss. Mainz, Abh. Math.-Naturw. Kl. 1, 1 (1961)
6. P. Jordan, J. Ehlers, R.K. Sachs, Gen. Rel. Grav. 45, 2691 (2013)
7. F.A.E. Pirani, Brandeis Lectures (Prentice Hall, New Jersey, 1965), p. 249
8. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, Oxford,
1983)
9. B.S. DeWitt, R.W. Brehme, Ann. Phys. 9, 220 (1960)
10. E. Poisson, A. Pound, I. Vega, Living Rev. Relativ. 14(7) (2011)
11. W.G. Dixon, Nuovo Cimento 34, 317 (1964)