G


T

*Partially based on an article by Baez and Bunn, AJP 73, 2005, 644

Overview




Einstein’s Equation: Gravity = Curvature of Space
What Does Einstein’s Equation Mean?
Needs Full Tensor Analysis
Consequences
 Tidal Forces and Gravitational Waves


Gravitational Collapse
Big Bang Cosmology … and more!


Stress and Curvature Tensors
What Have We Learned?

Preliminaries
 Special Relativity
 No absolute velocities,

Only relative
 Described by 4-vectors
 Depends on inertial coordinate systems

Field of clocks at rest with respect to each other

Preliminaries
 General Relativity
 Not even relative velocities

Except for two particles at same point
 Compare vectors by moving to same point



Need effects of parallel transport
On curved spacetime – path dependent
Einstein’s Equation
 Relative acceleration of nearby test particles in free fall

Einstein’s Equation – “Plain English”
 Consider small round ball of test particles rel. at rest
 Volume V(t), t – proper time for center particle
 In free fall it becomes an ellipsoid
 relative velocity starts out zero => 2nd order in time
V
V t

0
 
1
2





 flow of t
 t flow of flow of y
 y flow of x z

 x z






Summary of Einstein’s Equation



Flows – diagonal elements of T

P x
= Flow of momentum in x direction = pressure r
= Flow of t -momentum in tdirection = energy density
V
V t

0
 
1
2
 r 
P x

P y

P z

“Given a small ball of freely falling test particles initially at rest with respect to each other, the rate at which it begins to shrink is proportional to its volume times: the energy density at the center of the ball, plus the pressure in the x direction at that point, plus the pressure in the y direction, plus the pressure in the z direction.”

Consequences
 Gravitational Waves
 Gravitational Collapse
 The Big Bang
 Newton’s Inverse Square Law a
 
GM r
2

Tidal Forces and Gravitational Waves
 Test particle ball initially at rest in a vacuum
 No energy density or pressure
V t

0

0
 But curvature still distorts ball

Vertical Stretching


Horizontal Squashing
“Tidal forces”
 Gravitational Waves
 Space-time can be curved in vacuum
 Heavy objects wiggle => ripples of curvature
 Also produce stretching and squashing

Gravitational Collapse
 Typically, pressure terms small
 Reinsert units: c = 1 and 8 p
G = 1
 P dominates => neutron stars
 Above 2 solar masses => black holes
V
V t

0
 
4 p
G r m

1 c
2

P x

P y

P z


The Big Bang





Homogeneous and Isotropic
Expanding
Assume observer at center of ball of test particles.
Ball expands with universe, R(t)
Introduce second ball – r(t) r (0)

0 r (0)

R (0)
V
V t

0

3 r r t

0
 
1
2
 r 
3 P


Equation for R


Equivalence Principle – “at any given location particles in free fall do not accelerate with respect to each other”
So, replace r with R.
3 R
R t

0
 
1
2
 r 
3 P

3 R
R
 
1
2
 r 
3 P

 Nothing special about t=0.
 Assume pressureless matter
3 R
R
 
1
2 r
 Universe mainly galaxies – density proportional to R -3
Get Newtonian Gravity!
R
  k
6 R
2

Cosmological Constant




Last model inaccurate
Pressure of radiation important
Expansion of universe is accelerating!
Need to add
L
3 R
R
 
1
2
 r 
3 P 2
 3 R
R
 
2 k
R
3
 L
 L>0 leads to exponential expansion

Newton’s Inverse Square Law
 Consider planet with mass M and radius R, uniform density
 Assume weak gravitational effects R>>M, neglect P
 Consider
 Sphere S of radius r >R centered on planet
 Fill with test particles, initially at rest
 Apply to infinitesimal sphere
(green) within S
V
V t

0
 
1
2 r
V
V t

0

S



Inverse Square Law (cont’d)


The whole sphere of particles shrinks
Green spheres shrink by same fraction

V
V

1
2
 
 
 t
2  
1
4 r  t
2

V
S
 
V
P

 
V

 V 
V
P
 
1
4 r 
2 t V
P
 
1
4
2

V
S

4 p  r
 r
 
M
16 p r
2
 t
2
,
 r

1
2

2 a
 
M
8 p r
2
 
GM r
2 r

Mathematical Details
 Parallel Transport
 Measuring Curvature
 Riemann Curvature Tensor
 Geodesic Deviation
 Stress Tensor
 Connection to Curvature

Parallel Transport
Vector fields are parallel transported along curves, while mantaining a constant angle with the tangent vector www.to.infn.it/~fre

Flat and Curved Spaces
In a flat space, transported vectors are not rotated.
In a curved space they are rotated: www.to.infn.it/~fre

Measuring Curvature
Parallel Transport
Leading to Riemann Curvature Tensor
 lim

0 w
2
 w
1
( , )


2


R
   
 u v w

Compute Relative Acceleration
Consider two nearby particles in free fall starting at “rest”.
Particles are at points p and q.
Relative velocity.
Moving particles are later at p’ and q’.
Compute relative acceleration using parallel transport.
a

( v
2
 v
1

)

Relative Acceleration


Geodesic Deviation Equation

 lim

0 lim

0 a
 a



 lim

 
0
( v
2


2 v
1
)

R
   
 v u v
( , )
 
( , )
Second Derivative of Volume t lim

0 j j j
1
2 j a t
2 
 
 t lim

0 a
 j
  j
 
 
R v v
 
R tjt j
V
V
  j r j r j as t
 
V lim

0
V
V
 
R
  
R tt
Thus, Ricci => how volume of ball of freely falling particles starts to change.
(Weyl Tensor describes tidal forces and gravitational waves.)

What is R tt
?
Einstein Equation G


T
 where or
G


R


1
2 g R

R


T


1
2 g T

Thus, in every LIF for every point
Or,
R tt

T tt

1
2 tt

  tt

1 g T T T
2

 
1
2

T tt

T xx

T yy

T zz

V
V t

0
 
1
2
 r 
P x

P y

P z


Tensor Formulation – Flat Space
 Stress Tensor – for a continuous distribution of matter – perfect fluid
(density, pressure)
T
 
 r  p c
2
  u u
 p
 
 Symmetric T
 
T

 4-momentum density
 Signature Note:
T u

 c
2 r  p c
2 u
  pu
  c
2 r u
 u u

 c
2
, u
  
( , )

Stress Tensor Properties
 Divergence free

T
,


0
( r 
)
,

  r   u u u u
,

 p c
2 u
 
,
 u
 p c
2
  u u
,

 
2

  c p u u
 p
,

  
0


Continuity Equation
 Newtonian limit (small v , p )
Equation of Motion
 
,
 r  p c
2
 p c
2 u

,
 u
 
,
 u

0


 
 r
 t
  
( r v )

0
 
2
  c u u
 p
,


Newtonian Limit, Euler’s
Equation for perfect fluid r

  t
 
 v
  p

Tensor Formulation – Curved Space
 Fluid particles pushed off geodesics by pressure gradient r  p c
2 u
 
,
 u


   
2
  c u u
 p
,
 u

,
 u
 



 x
 dx d

 
 dx d


 d x d

2
 r  p
 2 d x

2 c d

2


   
2
  c u u
 p
,

Start with continuity and equation of motion to claim divergence free


Leads to more general formulation
Need Covariant Derivatives T
;



T


T

 x


 r  p c
2
  

T



  u u
 pg

  

T
 
0

Connection to Curvature
 Einstein’s attempts g
  kT

R
  kT

R
 
1
2
Rg
  kT

R
 
1
2
Rg
  L g
  kT


Connection to Metric
R

r

1
2 g


 x

 r



 g

 x


 x

 r

 g
 x



 g

 x



      
 
R


R


R

R

   g R

 g g

  


What Have You Learned?
 Special Relativity
 Space and Time
 General Relativity
 Metrics and Line Elements
 Geodesics
 Classic Tests
 Gravitational Waves


Cosmological Models
Einstein’s Equation
 Gravity = Curvature
 What Next?