Lecture 15
Schwarzschild equations of motion
Objectives:
• Planetary motion, start.
Reading: Schutz, 11; Hobson 9; Rindler 11.
15.1
Equations of motion
Writing µ = GM/c 2 , the Schwarzschild metric becomes ds 2 = c 2
µ
1 −
2 µ ¶ r dt 2
−
µ
1 −
2 µ ¶ −
1 r dr 2
− r 2 ¡ dθ 2 + sin 2 θ dφ 2 ¢ , and the corresponding Lagrangian is
L = c 2
µ
1 −
2 µ ¶ t ˙ 2 r
−
µ
1 −
2 µ ¶
−
1 r ˙ 2 r
− r 2
³
θ
˙
2 + sin 2 θ φ
˙
2
´
.
There is no explicit dependence on either t or φ , and thus ∂L/∂ t ˙ and ∂L/∂ φ
˙ are constants of motion, i.e
µ
1 −
2 µ ¶ t
˙
= k, r r 2 sin 2 θ φ
˙
= h, where k and h are constants.
h is the GR equivalent of angular momentum per unit mass.
60

LECTURE 15. SCHWARZSCHILD EQUATIONS OF MOTION 61
For k , recall that for “ignorable coordinates” such at t and φ , the corresponding covariant velocity is conserved , i.e.
˙
0
= g
0 β
˙
β
= g
00
˙ 0 = constant , where the third term follows from diagonal metric. Now x 0 g
00
= 1 − 2 µ/r , so
˙
0
=
µ
1 −
2 µ ¶ c t ˙ = kc.
r
= ct , while
Now p
0
= m ˙
0
, where p
0 is the time component of the four-momentum, and in flat spacetime p
0
= E/c where E is the energy, so
E = p
0 c = ˙
0 mc = kmc 2 , is the total energy for motion in a Schwarzschild metric.
NB k can be < 1, because in Newtonian terms it contains potential energy as well as kinetic and rest mass energy.
For the r component we have d dλ
µ ∂L ¶
∂ r ˙
−
∂L
∂r
= 0 , which gives d dλ
Ã
−
µ
1 −
2 µ ¶
−
1 r
!
−
Ã
2 µc 2 r 2 t
˙ 2 +
µ
1 −
2 µ r
¶
−
2
2 µ r ˙ 2 r 2
− 2 r
³
θ
˙
2 + sin 2 θ φ
˙
2
´
!
.
while the θ component leads to: d dλ
³
− 2 r θ
˙ ´
−
³
− 2 r 2 sin θ cos θ φ
˙
2
´
= 0 .
The last equation is satisfied for θ = π/ 2, i.e. motion in the equatorial plane.
By symmetry, we need not consider any other case, leaving
µ
1 −
2 µ ¶ −
1
¨ + r
µc 2 t ˙ 2 r 2
−
µ
1 −
µ
1 −
2 µ ¶ t ˙ = k, r
2 µ ¶ −
2 r
µ r 2 r ˙ 2
− r φ
˙ 2 = 0 r 2 φ
˙
= h.
,
For circular motion, ˙ r = 0, the second equation reduces to
µc 2 t ˙ 2 r 2
= r φ
˙
2 , and defining ω
φ
= dφ/dt and remembering µ = GM/c 2 , we get
ω 2
φ
=
GM
, r 3
Kepler’s third law! . . . somewhat luckily because of the choice of r and t .

LECTURE 15. SCHWARZSCHILD EQUATIONS OF MOTION
15.2
An easier approach
62
Rather than use the radial equation above, it is easier to use another constant of geodesic motion:
~
·
~
= g
αβ
˙
α x
β
= constant .
This is effectively a first integral which comes from the affine constraint, or, equivalently, from ∇
~
U = 0. It side-steps the ¨ term.
More specifically we have g
αβ x α x β = c 2 , for massive particles with λ = τ , and g
αβ x α x β = 0 , for photons.
15.3
Motion of massive particles
The equations to be solved in this case are thus c 2
µ
1 −
2 µ ¶ t ˙ 2 r
−
µ
1 −
µ
1 −
2 µ ¶ t ˙ = k, r
2 µ ¶ −
1 r ˙ 2 r
− r 2 φ
˙ 2 = c 2 r 2 φ
˙
= h.
,
Substituting for ˙ φ in the second equation and multiplying by − (1 −
2 µ/r ) gives r ˙ 2 + h 2 r 2
µ
1 −
2 µ ¶
− r
2 µc 2 r
= c 2 ¡ k 2
− 1 ¢ .
r 2 plus a function of r , “potential energy” equalling a constant.
Thus the motion in the radial coordinate is exactly equivalent to a particle moving in an effective potential V ( r ) where or, setting µ = GM/c 2 ,
V ( r ) = h 2
2 r 2
µ
1 −
2 µ ¶
− r
µc 2
, r
V ( r ) = h 2
2 r 2
µ
1 −
2 GM ¶
− c 2 r
GM
.
r

LECTURE 15. SCHWARZSCHILD EQUATIONS OF MOTION 63
One can learn much about Schwarzschild orbits from this potential.
The equivalent in Newtonian mechanics is easy to derive: r ˙ 2 + r 2 φ
˙
2
−
2 GM r
=
2 E
, m and r 2 φ
˙
= h . Thus so r ˙ 2 + h 2 r 2
−
2 GM r
=
2 E
, m
V
N
( r ) = h 2
2 r 2
−
GM
.
r
GR introduces an extra term in 1 /r 3 gravitational potential and 1 /r 2 in addition to the Newtionian 1
“centrifugal barrier” terms.
/r
15.4
Schwarzschild orbits
Three movies of orbits in Schwarzschild geometry were shown in the lecture.
Movies illustrate the following key differences between GR and Newtonian predictions:
• Apsidal precession of elliptical orbits
• Instability of close-in circular orbits
• Capture orbits