Transcript GR in a Nutshell

First Steps Towards a Theory of
Quantum Gravity
Mark Baumann
Dec 6, 2006
Where to Begin?
How might we quantize gravity?
Can we describe gravitational interaction by
exchange of a massless particle? If so, what kind of
particle?
Guess #1: Scalar (spin-0) particle
“Einstein-Fokker theory”
Problem: No gravitational bending of light or
precession of Mercury’s orbit
Guess #2: Vector particle, spin-1
Problem: Gravity exhibits repulsion
Where to Begin?
Guess #3: Vector particle, spin-2
The “graviton”
Perturbative approach: graviton is a perturbation to
a flat background spacetime, like a gravitational wave
Another approach:
Start with GR and modify it to make it look more like
quantum field theory.
General Relativity in a Nutshell
Einstein’s Field Equations:
R  12 Rg   8T
RHS: Matter/energy
LHS: Curvature of space
“Matter tells space how to
curve, and curved space tells
matter how to move.”
--J.A. Wheeler
Curvature of space is encapsulated in the “metric” g, which
tells us how far apart things are.
The metric is found by solving Einstein’s field equations.
General Relativity in a Nutshell
Metric in particle physics:
“Minkowski” metric that describes Cartesian space
and includes special relativity
 
1

 1





1


 1

Metric in GR:
g   something more complicate d
We said we were going to modify GR. How?
Spin and Torsion
•
•
•
•
•
Fundamental properties of a particle include: mass, spin
GR couples mass and the metric, a geometric property of
spacetime
Right now, GR is solely a macroscopic theory. However, for
a quantum theory of gravity, incorporating spin is
essential!
How could we couple spin to the geometry of spacetime?
• Idea: Couple spin to torsion
To understand torsion, we start with the covariant
derivative and the connection
Covariant Derivatives
Covariant Derivative in quantum field theory:
D     igA
“Converts” global gauge invariance into local gauge invariance.
Covariant Derivative in general relativity:
D     
λ
connection
The connection allows you to compare vectors from different tangent spaces.
The covariant derivative is defined so that it’s a tensor.
We give these the same name because they are, in fact, the same thing!
The GR version is more general.
Mathematical Roadmap
Riemannian
Manifold
Distance & Curvature
Physics done here
Add a metric
Covariant Derivatives
Add a connection
Manifold
Make it locally “flat”
Topological Space
Add a topology
Limits & Continuity
Set
More structure
More abstraction
Manifold with Connection
Torsion Defined
λ
Given a connection 
, torsion is defined as :
T    
λ
λ
λ
i.e. - it is a measure of the non-commutativity of the lower
two indices of the connection.
Torsion is a tensor quantity, unlike the connection.
Geometrically, the torsion measures how much rotation a
vector undergoes when you parallel transport it from one
tangent space to another.
Idea (due to Cartan, 1922): Couple spin and torsion
Connection
In classical GR, we solve the Einstein equations for the metric.
The standard choice for a connection in standard GR is the LeviCivita connection, sometimes called “Christoffel symbols.” We
get this connection if we start with a metric and assume the
connection is torsion-free (and also “metric-compatible”).
Now we are supposing the connection might not be torsion-free.
By allowing the possibility of torsion, we are “freeing” the
connection to be another variable. Whereas before we solved the
Einstein equations for the metric, now we wish to solve a new
set of equations for both the metric and the connection.
But which equations?
Einstein-Cartan Equations
Einstein’s equations are derived by varying the Einstein-Hilbert
action:
S EH   R  g d 4 x
V
where V is a 4-d spacetime volume and the Lagrangian = R,
which is the Ricci curvature scalar, which depends on the
connection.
 If the connection depends on torsion, then so does R.
This gives us the Einstein-Cartan action:
S EC
~
  R  gd 4x
V
Einstein-Cartan Equations
Varying the Einstein-Cartan action via the usual process, we get
the Einstein-Cartan equations:
Rμν  12 Rg   8T
Tμνλ  8S μνλ
The first equation has reproduced Einstein’s equations.
The second equation involves a spin density tensor S and a
modified torsion tensor T, and therefore couples spin with
torsion.
Coupling Spinors to GR
Quantum field theory uses a flat metric, but we don’t know how
to do QFT on a curved spacetime. So, why not make the metric
flat?
In GR, we can make the metric look flat at any point by a change
of basis. In other words:
 
  
g e e  
This new basis e is called the tetrad or vielbein basis, first
introduced by Weyl (1929).
Now we can proceed normally! However, what are the
consequences of this change of basis?
Tetrad Formulation
At every spacetime point, we have a different basis.
One consequence: the Dirac matrices are no longer constant, but
depend on position as follows:


  ea 
a
, where 
a
are theconstantmatrices.
 Every occurrence of  should be preceded by e
Once we find the Dirac matrices, we can compute the metric
through the usual equation:
1
2

 

  

 g

 We solve for the metric by solving for the  ‘s
Further Reading
Baez, J. and Muniain, J. Gauge Fields, Knots, and Gravity,
volume 4 of Series on Knots and Everything. World
Scientific, 1994.
-- Written by a mathematician, mathematically rigorous
but intuitively presented. Culminates with Ashtekar’s
New Variables.
Carroll, Sean M. Spacetime and Geometry: An Introduction to
General Relativity. Addison Wesley, 2003.
-- A mathematical introduction to GR. Good descriptions,
includes tetrad formulation.
Ortín, Tomás. Gravity and Strings. Cambridge University Press,
2004.