PHYM432 Relativity and Cosmology
14. Schwarzschild Spacetime Orbits
When looking at the metric
(dS) =
2
✓
1
2GM
c2 r
◆
(cdt)
dr2
2
(1
2GM
c2 r )
r2 d
2
r2 sin2 d⇥2
One would naturally like to think of Schwarzschild coordinates t as
the physical time as measured from a clock and r as the a physical
distance measured by a ruler, but this view would be wrong.
In GR, coordinates are arbitrary systems of markers used to
distinguish between events. If we want to know a distance an
observer would measure, we have to work out the relationship
between coordinate distances separating events and the distance
measured by a particular observer using the Metric of spacetime.
In GR coordinates have no immediate metrical significance.
Intervals of time and distance are measured by observers in a frame of
reference, and can change depending upon that frame. But the metric is
a tensor, and frame independent.
This gives great freedom in GR to choose different coordinate systems to
describe the same physical system. It can also lead to two apparently
unique solutions to Einstein’s field equation actually describing the same
thing.
Observers
Three local frames of reference are of particular use:
1) freely falling observer
- Gravity is “turned off” and space is locally flat
2) An observer in a fixed location around the Massive body
- a non-intertial force is needed to keep them in place
dr = d = d⇥ = 0
3) A distant observer far from the Mass - flat space
Gravitational time dilation
With the Schwarzschild metric, consider two events for a fixed observer
near the central Mass, r = rem , who emits two light pulses
dr = d = d⇥ = 0
(dS)2 =
✓
1
2GM
c2 rem
◆
(cdtem )2
dtem is the time between light
pulses
The proper time between the events is measured by a clock at rest at
the location of the events
2
(dS)
d 2=
2
✓ c
◆
d
d
2
em
em
=
=
1
✓
1
2GM
c2 rem
2GM
c2 rem
(dtem )2
◆1/2
(dtem )
The proper time is less than the coordinate time.
For a distant observer also at rest, the proper time will be
✓
◆1/2
2GM
d ob = 1
(dtob )
c2 rob
In the limit the distant observer is at infinity
d
1
= dtob
r!1
It turns out that the Schwarzschild coordinates dtob = dtem
So
d
d
1
1
= dtem and with
=⇣
d
1
em
2GM
c2 rem
d
em
=
✓
1
2GM
c2 rem
◆1/2
(dtem )
⌘1/2
r!1
For two light pulses emitted at rem The proper time between observing
these signals at infinity is larger than the proper time at the location of
emission near the Mass.
d em
d 1=⇣
⌘1/2
2GM
1 c2 rem
So if 1 second passes at r = rem
then for a distant observer at r !
⇣
1
1
2GM
c2 rem
⌘1/2
seconds will pass
1
Clocks run Slower in the gravitational well.
The effect of slowing the rate of a clock in a gravitational field is called
Gravitational time dilation
Notice both observers are still at
rest, but a time dilation occurs
anyway.
r!1
Gravitational Redshift
With d em being the time between successive “peaks” in a light wave
emitted at r = rem
then
d
1
= ⇣
d
1
em
2GM
c2 rem
⌘1/2
which is related to the frequency by
✓
◆1/2
2GM
f1 = fem 1
c2 rem
f1 =
1
d
1
and wavelength by
1
=⇣
em
1
2GM
c2 rem
⌘1/2
Which is the phenomena of Gravitational Redshift
Singularities
For the heaviest Neutron star known
Rs=6 km being twice as heavy as the sun.
The surface is ~10 km.
So the gravitational redshift is
1
1
=
em
1
= 1.6
6 1/2
10
em
Violet light (4000 ang) is at the blue-edge of what we can see, will be
shifted to (6400 ang) which is red, and on the red-edge of the visible
spectra our eyes can see.
Proper distance
The proper distance between two events at the same coordinate time is
dt=0
p
d =
(ds2 )
(dS) =
dr2
2
(1
2GM
c2 r )
r2 d
Along the same radial direction
(d ) =
2
r2 sin2 d⇥2
d = d⇥ = 0
dr
(1
2GM 1/2
c2 r )
This gives the relation between the arbitrary coordinate distance dr and
the real physical distance d
(d ) =
dr
(1
2GM 1/2
c2 r )
With the Schwarzschild coordinates, the length of the spacial coordinate
dr stays the same on the flat plane as you move to larger distances, but
the increased curvature “lengthens” the relative proper distance as you
move closer to the massive body. Of course d is the real physical
distance and dr is just the convenient coordinates we have chosen.
Geodesic motion in Schwarzschild Spacetime
With the Schwarzschild metric
(dS) =
2
✓
1
2GM
c2 r
◆
(cdt)
dr2
2
(1
2GM
c2 r )
r2 d
2
r2 sin2 d⇥2
We can use the geodesic equation to solve for the orbits of “test particles”
d2 X
0=m
+m
2
d
dX µ dX ⇥
µ⇥
d
d
↵
1
= g↵
2
✓
g
g
+
X
X
The affine connection parameters follow from the metric...
g
X
◆
The four equations of motion are...
In principle, we could solve for these four differential equations, and
fully describe the worldline (orbit) of our test particle.
However, we can use a much simpler approach, using conservation of
energy and momentum to simplify these equations.
In physics, conserved quantities such as energy are associated with
symmetries.
How does one tell if a spacetime has a symmetry?
If a transformation leaves the metric unchanged, there is a symmetry
with respect to the transformation.
We saw for the Schwarzschild metric
t ! t0 = t + t 0
x0 ! x0 + constant
it was time independent, with no t quantities in any of the metric
coefficients.
= (1, 0, 0, 0) lyes along the direction of the symmetry,
The unit vector
and is called the Killing vector (after Wilhelm Killing).
0
In an arbitrary coordinate system, a conserved quantity along a geodesic is
gµ
⇥·U
⇥ = constant
µ
U = constant
With a time independent metric
✓
(1)
0
= (1, 0, 0, 0)
the only non-zero term is
g00 0 U 0 = constant
1
✓
◆
cdt
(1)
= constant
d
◆✓ ◆
2GM
dt
E
=
c2 r
d
mc2
2GM
c2 r
1
◆
✓
So time independence is related to energy conservation, and the constant
turns out to be the energy per unit mass.
Another important killing vector is
l↵ = (0, 0, 0, 1)
gµ lµ U = constant
r sin2
U 3 = constant
J
2 d⇤
2
the angular momentum
=
(2) r sin
d⇥
m
We can now use these quantities with the scalar invariant quantity of the
test particles 4-velocity
c2 = g↵ U ↵ U
dx↵ dx
c = g↵
d d
There are four non-zero terms in the sum
2
Use coordinates where ang. momentum points along the polar axis
confining the motion to a plane ✓ = ⇡/2 so
d✓/dt = 0
Substituting (1) and (2) give
or
To simplify even further, we note that in the ✓ = ⇡/2 plane
2 d⇥
J
r
=
d
m
to put our equation into terms of dr/d
Then we can use a standard trick in orbital mechanics and introduce
1
u=
r
Differentiate with respect to phi, which will then give us the
orbital shape equation.
d2 u
GM m2
3GM u2
+u=
+
2
2
d
J
c2
Which is very similar to the corresponding Newtonian mechanics
equation.
d2 u
GM m2
+u=
2
d
J2
The extra term represents GRs contribution and introduces unique
differences with the orbits of particles.
1) The orbits rotate in the 90 degrees plane, causing precession of the
perihelion
2) there is a lower limit to the radius of a circular orbit
This can be seen if we identify the effective potential from
Vef f
J2
=
2m2 r2
✓
1
2GM
c2 r
◆
GM
r
Which differs from the Newtonian case
Vef f
J2
=
2m2 r2
GM
r
A
B
In Newtonian mechanics there is one stable circular orbit, in GR there are
two, one unstable point A and one stable B
d2 u
=0
2
d
3GM u2
c2
circular orbit
GM m2
u+
=0
J2
2
c
u=
±
6GM
r
12GM 2 m2
J =
c2
2
1
12G2 M 2
c2 J 2
stable circular orbit
1
c2
=
r
6GM
r = 3Rs
scattering
orbit
Plunge
Orbit
Precession
Orbit
Lamborurne
5.2, 5.3, 5.5, 5.6a, 5.6b