General Relativity
Outline
• Gravitation and Electromagnetism Compared
• The Equivalence Principle & Inertial Frames
• Gravitation and Geometry
• Gravitation and Electromagnetism Compared Part Deux
• Solutions (SR, Black Holes, Cosmology)
• Gravitation and Electromagnetism Combined
Gravitation and Electrodynamics
Compared
Coulomb' s Law


1 q1q2
Fq1q2 =
rˆ12 = q2 Eq1 (r2 )
2
4πε o r12
Newton' s Law of Gravitation


m1m2
Fm1m2 = −G 2 rˆ12 = m2 g m1 (r2 )
r12
Newton' s 2nd Law


Fq1q2 = m2 a2
Newton' s 2nd Law


Fm1m2 = m2 a2

q2 
am2 ,q2 (r2 ) =
Eq1 (r2 )
m2

m2 
am2 (r2 ) =
g m1 (r2 )
m2
The Equivalence Principle
If minertial=mgravitational for all objects, then on small enough scales*
gravitation and acceleration are interchangeable.

a = g ˆj

g = 0 ˆj
light
light

a = 0 ˆj

g = − g ˆj
Starting from a uniformaly accelerating situation where we know that light’s
trajectory will “bend”, we can immediately infer that gravity bends light trajectories!
*By small enough, we mean such that the gravitational field is approximately uniform, i.e. no tidal effects.
“Inertial” Frames
The equivalence principle has deep consequences for Newton’s 1st law.
To establish an inertial frame, we need a particle which is experiencing no force.

E
E&M:
q=e
-defines an inertial reference frame since it is
feeling no force and hence has no acceleration.
q=0

g
Gravity:
mass
m = me
m=0
Since all objects (even massless ones) undergo
acceleration in the presence of gravity, it is
impossible to establish an inertial reference frame.
Gravitation and Geometry
Einstein thought long and hard about this
“universal” quality of gravity. It is unlike
any other force in nature in that every
particle reacts the exact same way to it.
He realized that one “thing” which influences
all particles in the same way is the geometry
of spacetime itself.
It had hitherto been assumed that the
geometry was trivial, e.g. “flat”, but Einstein
began to think of what might happen if the
geometry was curved.
From an Idea to a Formalism
Galilean Relativity:
 1 0 0  ∆x 

 
(∆x ∆y ∆z ) 0 1 0  ∆y  = ∆x 2 + ∆y 2 + ∆z 2 = constant ∆t 2 = constant
 0 0 1  ∆z 

 
Special Relativity:
 −1

0
(c∆t ∆x ∆y ∆z )
0

0

0
1
0
0
0
0
1
0
0  c∆t 


0  ∆x 
2
2
2
2
2
=
−
∆
+
∆
+
∆
+
∆
= constant
c
t
x
y
z



0 ∆y




1  ∆z 
General Relativity:
 g tt

g
(c∆t ∆x ∆y ∆z ) xt
g
 yt
g
 zt
g tx
g ty
g xx
g yx
g zx
g xy
g yy
g zy
g tz  c∆t 


g xz  ∆x 
= constant



g yz ∆y




g zz  ∆z 
g ij = g ij (t , x, y, z )
Dynamical Geometry
The cornerstone
 of General Relativity is to allow
the metric g (t , x, y, z ) to be a dynamical field
whose value is determined by the distributions of
mass and energy that are present.
This is similar
to having

 the electric and magnetic
fields E (t , x, y, z ) , B (t , x, y, z ) determined by
the distribution and motion of charges that are
present.
In fact…...
Gravitation and Electrodynamics
Compared Part Deux
E&M
  ρ
∇⋅E =
 
∇⋅B = 0
εo

 
∇ × E = − ∂B

 

∇ × B = µ o J + µ oε o ∂E
Describes how you create
“fields” from sources.
GR
∂t
 1

R − gR = 8πGN T
2
∂t

  
F = q( E + v × B)


F = ma
Describes how particles
Respond to the “fields”.
b
c
d 2 xa
dx
dx
a
+
Γ
=0
bc
2
ds
ds ds
Solutions
 

R − gR / 2 = 8πGN T are 10 coupled, nonlinear, hyperbolic-elliptic partial differential equations.
Exact solutions are crazy hard to find, unless one uses a lot of symmetry!
Time-independent
with Maximal Symmetry :

2
2
2
2
2
2
T is 0 everywhere ⇒ ds = −c ∆t + ∆x + ∆y + ∆z
describes the empty space of Special Relativity!
So the connection between Special and General Relativity is simply that SR is one many
solutions of GR. In particular it is the most trivial one, i.e. without any sources present
and hence no gravity!
Solutions
 

R − gR / 2 = 8πGN T are 10 coupled, nonlinear, hyperbolic-elliptic partial differential equations.
Exact solutions are crazy hard to find, unless one uses a lot of symmetry!
Time-independent
with Spherical Symmetry :

T is spherically symmetric ⇒ ds 2 = −(1 − 2GM / c 2 r )c 2 ∆t 2 + (1 − 2GM / c 2 r ) −1 ∆r 2 + r 2 dΩ 2
describes exterior stellar geometry which is useful for GPS and orbit calculations as well
as connecting GR to Newtonian gravity.
Taken as an interior solution as well, this metric
describes a Shwarzchild Black Hole.
More complicated variations of this metric
describe spinning and/or charged black holes
which have considerably more exotic properties.
Solutions
 

R − gR / 2 = 8πGN T are 10 coupled, nonlinear, hyperbolic-elliptic partial differential equations.
Exact solutions are crazy hard to find, unless one uses a lot of symmetry!
Time-dependent
with Homogeneity and Isotropy Symmetry :

T is that of a perfect fluid ⇒ ds 2 = −c 2 ∆t 2 + a(t ) 2 (dr 2 / 1 − κr 2 ) + a(t ) 2 r 2 dΩ 2
describes a Friedmann-Robertson-Walker cosmology with different possible values
of the spatial curvature depending on the matter-energy content of the universe.
Based on the observed content of the
universe, this metric also predicts an
initial Big-Bang singularity.
Gravitation and Electrodynamics
Combined
The similarity between E&M and GR goes even deeper.
The Kaluza-Klein idea is that pure GR in higher
dimensions can be reduced to 4D GR plus extra “stuff”
when the extra dimensions are small.
When this idea is applied to GR in 5D where the 5th
dimension is a small circle, we get 4D GR and E&M!!!
Makes you wonder what could be done with say .... 10 dimensions?!