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PH30101 General Relativity
Prof Tim Birks
“General relativity without tensors”
General relativity (GR) states that spacetime is deformed by
gravitating masses. Freely-moving objects follow straight lines
(or their nearest equivalents) in this curved spacetime, even in
the presence of gravity. GR is our current theory of gravity and,
alongside the "standard model" of quantum/particle physics,
forms our best account yet of how the Universe fundamentally
works.
A complete treatment of GR relies on tensor analysis, a chunk of
advanced mathematics we'd need to spend a lot of time learning
before even starting the actual physics. But, knowing the
curvature of spacetime, we can deduce the motion of particles
and light without knowing about tensors. (It's still mathematical,
but it's maths you already know.) In this unit we will use this
approach to examine what curved spacetime means, compare the
predictions of GR and Newtonian gravity, and explore the
properties of the quintessential GR object: the black hole.
Spacetime curvature is described by metrics , which we won't be
able to derive without tensors. We'll just take them as given. But,
when you learned quantum mechanics, did it bother you (did
you even notice) that you never saw where Schrödinger's
equation came from?
TA Birks, University of Bath, 10 January 2023
0. Course admin
Revision : You will need material from previous units, including:
special relativity (SR): PH10103 for Physics students, PH
for Maths and Physics students, or PH20106/PH20114 for
students who transferred to Physics from other courses.
Newtonian mechanics and gravity: including gravitational
potential, angular momentum, orbits, planetary motion, impact
parameter
calculus: differentiation and integration (including line and
multiple integrals), polar plots, ordinary differential equations
(separable, forced s.h.m.) and especially coordinate systems like
spherical polars. But, no vector analysis or complex numbers.
geometry: basic stuff (triangles, circles, parallel lines), curves
(ellipses, hyperbolae) and spheres (surface area, latitude and
longitude, great circles).
thermal physics and quantum mechanics: entropy, microstates,
black-body radiation, the uncertainty principle(s).
I don't schedule office hours, but I'm usually happy to be
interrupted. To contact me outside timetabled contact time, use
email ([email protected]) or visit my office (8W 4.13) if the
door is open. It's OK for students to get in touch and ask me
questions!
0. Course admin
Structure of the unit
GR: curved
spacetime
equivalence
principle
Newton's
gravity
special
relativity
acceleration
GR: the geodesic
equation
Schwarzschild
metric
orbits of particles
and light
3 classical
tests
black holes
Kerr metric
thermodynamics and
quantum gravity
optional
lecture
non-Euclidean geometry
L1,
L
L
L
L
L1- 6
L7,
L
L10- 12
L3- 12
L13- 16
L17- 18
L19- 20 L
0. Course admin
Table of contents
- 0 Course admin section title page lecture
- 1 Introduction: deforming time and space 6 L
- We needed a new theory of gravity!
- The principle of equivalence
- Consequences of the principle of equivalence 12 L
- General relativity
- 2 Newtonian gravity 18 L
- Time
- The effective potential
- Shapes of orbits
- 3 Special relativity 26 L
- Relativistic units
- Spacetime
- The interval d s^2 and the Minkowski metric
- Acceleration in SR 34 L
- 4 Geometry 41 L
- Flat space
- Curved space
- Flat spacetime
- Curved spacetime
- 5 The Schwarzschild metric 50 L
- What the Sch. coordinates mean
- Gravitational time dilation 56 L
- 6 The geodesic equation of motion 60 L
- 7 Orbits in Schwarzschild spacetime 65 L
- Equations of motion
- The effective potential
- Bound orbits 71 L
- Radial (vertical) motion
- Photon orbits 77 L
- 8 Schwarzschild black holes 83 L
- The singularity at the Schwarzschild radius
- Painlevé-Gullstrand (PG) coordinates 88 L
- The central singularity ( r = 0) 93 L
- Kruskal-Szekeres (KS) coords 98 L
- 9 Kerr (rotating) black holes 105 L
- The Kerr metric
- Orbits around Kerr black holes 110 L
- 10 GR and quantum mechanics 116 L - The thermodynamics of black holes - Quantum gravity 120 L
These two viewpoints contradict one another: m can't both move
upwards (and eventually collide with the upper stream) and not
do so. This exemplifies the fact that:
In our thought experiment, F is balanced by a new repulsive
force between co-moving masses called "gravito-magnetism", cf
the well-known velocity-dependent relativistic force between
electric charges. But rather than patching up Newton's theory in
this way, Einstein preferred to start from scratch with:
Then look at things in frame S' moving at speed v to the right
relative to S. Now the lower stream is at rest, m moves to the left
at speed v , and the upper stream moves to the left at a speed
greater than v.
Lecture 1^7
According to SR, moving objects undergo length contraction.
The upper stream moves faster in S' than in S and experiences
more length contraction, so its spacing is < l. The lower stream
moves slower in S' than in S and experiences less length
contraction, so its spacing is > l.
There's therefore more mass (per unit length) above m than
below, and a net upward gravitational force F on m.
m accelerates upwards
Newtonian gravity is not consistent with SR.
1. Introduction / A new theory of gravity
Lecture 1^8
Why are they identical?
Actually there are other forces that accelerate independently of
mass. For example, "g-forces" that push you backwards in an
accelerating rollercoaster, centrifugal forces that pull outward on
a curved path and Coriolis forces that spin weather systems.
What all these forces have in common is that they don't exist...
They are pseudo-forces that appear only in accelerating (ie, non-
inertial) frames of reference. The acceleration that all masses
seem to have in common is merely the acceleration of the frame
itself.
2
Mm mg G r
=^ [ ma^ =^ F ]
inertial mass m
(resistance to
acceleration)
gravitational mass m
(source of
gravitational force)
The principle of equivalence
Einstein's thinking on gravity was based on a familiar result
from Newton's theory - the acceleration g of test mass m due to
mass M is independent on m :
The fact that all free-falling masses accelerate equally was well
known before Newton (Galileo etc) and has been experimentally
verified to within one part in 10
12
. Yet in Newton's theory it is an
astonishing coincidence, because the m 's on both sides of the
above equation represent logically-distinct concepts:
1. Introduction / Principle of equivalence
The word "local" is important. Unlike uniform acceleration,
gravity has a centre. Free-falling frames have vector
Lecture 1^10
There are no universal inertial frames, only local ones.
flat Minkowski spacetime
of SR into the curved
spacetime of GR. (Like
the way stitching
together lots of flat city-
scale maps produces a
spherical surface on a
continental scale.)
Cut out the U-shape of stitched-together small-scale maps and lay the two maps for the
north pole over each other (with the right orientation). The paper strip forms part of a globe, even though each individual map is approximately flat.
accelerations that vary from
place to place, in magnitude
( g is slightly bigger at your
feet than your head) and
direction (the vertical in Bath is
~1½º away from the vertical in
London). A big-enough
experiment can use this spatial
variation to distinguish gravity
and accelerated motion, and we
can't find a common frame of
reference that eliminates gravity
everywhere:
Stitching together neighbouring local inertial frames across an
extended region turns the
1. Introduction / Principle of equivalence
The essence of gravity in GR - what can't be eliminated by
moving to a new frame of reference - is the spatial variation left
over when the gross effect of gravity is subtracted by moving to
a freely-falling local inertial frame.
Imagine a free-falling sphere of loose gravel, ignoring air
resistance etc. The bottom of the sphere has bigger g than the
top, and at the sides the directions of g converge slightly to point
to the centre of the gravitating mass
. Shifting to the (inertial)
frame of the centre of the sphere means subtracting gaverage :
Lecture 1^11
What's left over is a vertical tension and horizontal compression,
tending to deform the sphere into an ellipsoid. If the gravel was
water, with a rocky ball inside rotating once per day, the whole
lot in free fall towards the Moon, you might recognise these left-
over forces as the tides.
Tidal forces are the essence of gravity.
They encapsulate the spatial variations discussed on the previous
page. In our unit we won't study tidal forces much, but this
concept is central to the tensor formulation of GR as a whole.
* The gravel doesn't have to actually hit the gravitating mass. Free fall is just motion without forces other than gravity, and can be upwards or sideways (like an orbit) as
well as the classic vertical drop.
1. Introduction / Principle of equivalence
Here's a time-line for these various events:
Lecture 2^13
So D tB < D tA : the pulses arrive at B more often than they leave A.
Nothing paradoxical so far: the times of flight of the pulses are
clearly different, just giving a fancy kind of Doppler shift.
But the equivalence principle says we get the same result in a
room R' at rest in a gravitational field g.
B A 2 1 A 1 2 A 1 1 2 A
gh gh t t t t t t t t t c c
D = D + - = D + - D - = - D
Now the room doesn't move. The times of flight are now the
same. But the equivalence principle demands D tB < D tA still. An
inhabitant of the room, measuring these times, concludes:
Gravitational time dilation: time passes more slowly
lower down in a gravitational field*.
The time D tB between the two pulses reaching B is therefore
* The difference is tiny on Earth: ~1 m s per century between a typical ceiling and floor.
1. Introduction / Gravitational time dilation
• Consequence #2: curved spacetime
In a ( t versus x ) spacetime diagram*^ of the events in R', A and B
are at rest so their worldlines are vertical, with constant values of
x separated by h. This means lines CD and EF are parallel.
The two pulses of light travel at the same speed c , so their
worldlines make the same angle to the axes. This means lines
CE and DF are also parallel. By definition, the quadrilateral
CDFE is therefore a parallelogram.
Lecture 2^14
But CD = D tA and EF = D tB , so the time dilation result means
CD EF
CDFE is a parallelogram with a pair of unequal opposite sides!
Obviously this contradicts a basic theorem of plane geometry as
developed by the ancient Greeks, such as Euclid. In fact such a
shape cannot be accurately drawn on a flat sheet like the above
diagram. The sheet would need to be curved.
Gravity causes spacetime to be curved.
The geometry of spacetime is "non-Euclidean".
1. Introduction / Curved spacetime
* x is the height coord: it's a shame spacetime diagrams always put space horizontally!
Ordinary mass (represented by energy density) is merely T^00 , the
time-time component of T
m
. Other sources of gravity in GR are
momentum density (space-time components) and pressure and
stress (space-space components). Meanwhile, on the LHS, G
m is
a set of complicated derivatives of something called the metric ,
which describes the spacetime curvature geometrically.
The use of tensors is an elegant way to express the principle of
general covariance , which states that the laws of physics should
be valid in all frames of reference, not just inertial ones. But
written out in full, the Einstein equation becomes 10 coupled
nonlinear partial differential equations in non-cartesian
coordinates. Actually solving these equations, to get the metric,
is hideously complicated. Indeed, it has only ever been done
exactly in a handful of very simple cases. We'd need to spend
most of the semester learning tensor analysis before even
beginning any physics. Therefore, I regret to inform you that
Lecture 2^16
In this unit*, we will not learn about how "matter
tells spacetime how to curve".
* To study GR with tensors, and learn about matter telling spacetime how to curve, take PH40112 Relativistic Cosmology.
1. Introduction / General relativity
It relates a small spacetime interval ds^2 (as in SR) to small
changes in four coordinates. The coordinates resemble spherical
polars ( r , q, f) with time t tacked on. The metric generalises the
Pythagoras theorem, and is a geometrical description of how the
mass warps spacetime in its vicinity.
Note there are no tensors to be seen, just ordinary calculus. You
don't need tensors to perform calculations with it either. Tensors
are elegant but for part they are optional - and we will opt
out
! I am therefore pleased to inform you that
Lecture 2^17
In this unit, we will take spacetime metrics derived
elsewhere to learn about how "spacetime tells matter
how to move".
1 2 2 2 2 2 2 2 2 2 2 2
1 1 sin
GM GM
ds c dt dr r d r d c r c r
q q f
= - (^) - (^) + (^) - (^) + +
Part of Wheeler's comment means that the motion of particles
(and light) is determined by the metric found in part . The rule
is that SR remains valid locally, and that free-falling particles
follow geodesic world-lines. (In a curved space or spacetime, a
geodesic is the nearest thing to a straight line.) This plays the
role of an equation of motion, like F = ma , telling us how the
particle moves in a given spacetime.
Although the metric is really a tensor, it is actually possible to
write it as a so-called "line element" without knowing anything
about tensors. Here's an important example, the Schwarzschild
metric for a spherically-symmetric mass M :
* Other omitted GR topics: for cosmology, gravitational waves and other astrophysics,
see PH40112 Relativistic Cosmology and PH40113 High Energy Astrophysics; for
exotic matter and warp drives - get back to me when they've become science.
1. Introduction / General relativity
Lecture 3^19
The effective potential
A test particle m moves near a gravitating point-mass M at O.
2 d L mr mvb dt
f
- angular version
of p = mv : I^
- moment of
momentum:
p × ⊥ distance
For central forces like gravity, angular momentum is conserved.
The size of m is rarely important, so we'll work with angular
momentum per unit mass or specific angular momentum l :
L (^) 2 d l r vb m dt
f
= = =^ (i)
• The angular momentum of m , defined by L = r × p , has a
magnitude L that can be thought of in two ways [revision!]:
• The energy of m = KE + PE
GMm mv r
Likewise, energy per unit mass or specific energy EN. If we write
speed v in its radial and azimuthal*^ components vr and v f:
2 2 (^1 )
N
dr d GM E r dt dt r
f
vr = dr / dt v f = r
= radial KE/ m + azimuthal KE/ m + grav. potential
(N for
"Newtonian")
[ v
2 = ( vr )
2
2 ]
* "azimuthal" = in the tangential direction, ie perpendicular to the radial direction
2. Newtonian gravity / Effective potential
Lecture 3^20
This is the energy equation for a particle undergoing 2 - D orbital
motion. But, if we substitute d f/ dt = l / r^2 using the angular
momentum formula (i) we can pretend that the particle is
undergoing 1 - D radial motion, by combining the azimuthal KE
with the true potential to make an effective potential VN ( r ). The
key feature is that the form of VN depends only on position r not
velocity, which is what we expect from a potential function:
(^2 )
2
E N K N VN
dr GM l
dt r r
2
2
N
GM l V r r r
KN = radial KE/ m
effective potential
true gravitational potential "centrifugal term"
(from azimuthal KE)
Now we can analyse the radial part of the particle's motion as if
it was just moving along r subject to the effective potential VN.
[When we need the azimuthal part of the motion, we can solve
the angular momentum formula (i).]
Note that KN can never be negative - it's something squared. So
the particle can only be where EN VN ( r ), and the gap between
EN and VN relates to the particle's radial ( r direction) speed there.
A plot of VN ( r ) therefore tells us a lot about the possible orbits.
(ii)
(iii)
2. Newtonian gravity / Effective potential
