1. General Relativity Physics Honours 2009
Florian Girelli
Rm 364, A28
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2. Instead they will be on the 23rd September, from 10am to 12pm
Courses on the 16th September are cancelled due to a university strike.
Instead they will be on the 23rd September, from 10am to 12pm
The third assignment will be given on the 21st September (it was initially planed to be given the 16th).
It should be handed out the 5th October.
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3. Einstein eq. and applications
Up to now, you have seen:
Space-time defined by a metric, (global) symmetries given by Killing vectors.
Propagation of matter (particle) given by geodesic equation
What is the analog of the Poisson equation?
Geometry tells how matter should propagate. Matter tells how geometry should curve.
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4. Source of gravity = mass density =M/V ~ energy/volume.
needs generalization to the 4d case.
3d volume:
4d volume:
3d volume embedded in 4d:
Energy momentum
To obtain a momentum from a 3d volume embedded in 4d, we need a tensor encoding the “momentum density”.
Stress-energy-momentum tensor
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5. Stress-Energy-Momentum
To understand what this means, consider flat spacetime at a constant time; this is a 3-d space with n=(1,0,0,0).
energy density
momentum density
stress tensor: measures internal forces that part
of the medium exert on the others (Cauchy 1822)
force per unit of area ~ pressure
Chapter 22
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6. Examples Fluid: If we are at rest with respect to a perfect fluid then
In flat spacetime, we can extend this to a moving fluid so
It should be clear that in the rest frame, this becomes the above rest frame expression.
Particle: it is not difficult to see that we have
From a lagrangian:
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7. Gravity as curvature
Einstein’s key idea is to understand that gravity is encoded in the geometric properties of spacetime: curvature.
Define the parallell transport of vectors
Define curvature
Check that curvature is related to the gravitational potential
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8. Covariant Derivative
We now need to look at the mathematical structure behind general relativity. This begins with the concept of the covariant derivative. Let’s start with flat, Minkowski spacetime;
Where the second expression gives us the derivative of the vector in the t direction.
To compare the vectors at two points, we have had to parallel transport one vector back along the path to the other. In Cartesian coordinates, this is no problem as the components of the vector do not change. This is not true in general.
Chapter 20.4
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9. Covariant Derivative
Remember, in general curvilinear coordinates the basis vector change over the plane. This change of basis vectors needs to be considered when calculating the derivative.
Hence, the Christoffel symbol can be seen to represent a correction to the derivative due to the change in the basis vectors over the plane. For Cartesian, these are zero, but for polar coordinates, they are not.
A vector field v is parallely transported along t if
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10. Example: Geodesics
Imagine we have a straight line path though space time, parameterized by , and this path have a unit tangent vector u then
Hence, geodesics are paths that parallel transport their own tangent vector along them (i.e. there is no change to the tangent vector along the path).
Think about a straight line path through polar coordinates!
Geodesics in curved spacetime are just a generalization of the above.
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11. Example: Free Falling Frames
Orthonormal bases are parallel transported by definition
This allows to construct freely falling frames. If we have someone falling from infinity radially inwards in the Schwarzschild metric, then
Where the first component is the 4-velocity, but you should check for yourself that the other components are parallel transported.
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12. Covariant Derivatives for tensors
We can generalize the covariant derivative for general tensors
(This is straight forward to see if we remember that t = v w and remember the Leibniz’s rules).
What about downstairs (covariant) component?
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13. Covariant Derivative for tensors
One of the fundamental properties of the Christoffel symbols is
In Special Relativity, the stress-energy tensor is conserved.
This is naturally generalized to the curved case using the covariant
derivative:
This is not anymore properly a conservation of energy since if
spacetime is dynamical matter can exchange energy with it
(« local conservation of energy » cf Example 22.7)
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14. Introducing curvature
To measure curvature we analyze the parallell transport of the vector along a closed loop: curvature measures the failure of coming back to the same vector. Mathematically it can be obtained from
is the Riemann tensor.
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15. Riemann Tensor Writing this in a local inertial frame, it becomes
Leading to some immediate symmetries (true in general);
Instead of 256 independent components, this tensor really only has 20 (Phew!)
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16. Riemann Tensor Notation: Bianchi Identity (proof written in inertial
frame to neglect
Christoffel terms)
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17. Contractions
Contracting the Riemann tensor gives firstly the Ricci tensor
Contracting again gives the Ricci scalar
We also have the Kretschmann scalar, which is the measure of the underlying curvature
If this is not zero, the spacetime is not flat! For the Schwarzschild metric, we have
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18. Curvature and gravity
Curvature also measures the failure of initially parallell geodesic to remain parallell.
We consider two particles in a gravitational potential, first in the
Newtonian formalism, then in the GR case and compare.
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19. Newton deviation In the Newonian picture, we have the Newton equation.
If we consider two nearby particles separated by a vector 
Taking the leading terms, we get the Newtonian deviation between two trajectories
Chapter 21
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20. Geodesic Deviation
The separation of two free falling objects gives a measure of the underlying curvature. We need to consider the paths of two nearby geodesics, with a 4-space separation as a function of the proper time along both curves.
Where the Riemann Curvature tensor is
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21. Geodesic Deviation
This is easier to understand in the free falling frame. We need project the deviation vector into the orthonormal frame
Remembering what the 4-velocity is in the free falling frame, then
Where the Riemann tensor has been projected onto the orthonormal basis.
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22. Geodesic Deviation In the weak field limit
If we assume that our objects fall slowly (non-relativistic) along the geodesics, then the orthonormal and coordinate frame are approximately the same, so
We can calculate the Christoffel symbols from the metric and then calculate the components of the Riemann tensor.
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23. Gravity is therefore related to the geometry of spacetime,
Geodesic Deviation
Keeping only the lowest order terms, we find
In the weak field limit, non-relativistic limit, we recover the result from Newtonian physics.
Remember, the Riemann tensor is something that describes geometry, but here it is related to something physical, the gravitational potential.
Gravity is therefore related to the geometry of spacetime,
ie its curvature.
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24. Looking for Einstein Equation
We have now the tools to determine some equations of motion intertwining gravity and matter. The source should be expressed in terms of the stress energy tensor.
Between 1910 and 1913, Nordstrom proposed to consider generalizations of the Poisson equation (arXiv:gr-qc/ ):
Wrong for many reasons.Eg gravitiational
field carry energy and should self gravitate:
non-linear effects
Einstein and Fokker showed that this latter proposition can be put into a geometric
shape
This is the first geometrical candidate for gravity (scalar gravity).
Not physical: no bending of light (metric is conformally flat), wrong precession…
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25. Looking for Einstein Equation
Gravitational degrees of freedom are encoded by curvature (Riemann tensor, Ricci tensor, Ricci scalar), we look for the equations which give the Poisson equation in the Newtonian limit and are second order in derivatives in the metric.
Einstein first tried
But this fails to work since we have also the conservation of the stress energy tensor and
Instead we have (thanks to the Bianchi identities)
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26. Einstein Equation
We (finally) get the Einstein equation (for zero cosmological constant): matter tells how spacetime to curve.
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27. GR is not alone… Chapter 10.2
General Relativity is the « simplest » theory, but one can generate other theories compatible with experiments by adding scalar, vector fields, higher order contributions in curvature... Introduce Parametrized Post-Newtonian parameters: they determine how the gravitational theory candidate is different from Newtonian gravity (in the weak field limit). Experiments put constraints on the value of these coefficients. See for example arXiv:gr-qc/
Tiny modifications of Schwarzschild metric
Chapter 10.2
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28. GR is not alone…
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29. Lecture Notes 5

30. Example 20.8
Hence, we can have a general form for the acceleration of an object due to an applied force
In flat Minkowski space time (or local inertial frame) this is
So, what acceleration is required to hover at a distance r from a Schwarzschild black hole? The 4-velocity is given by
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31. Example 20.8 What acceleration do you need to remain at r?
As the 4-velocity is independent of time, we find
Only non-zero Christoffel symbol is
As expected, the force is in the radial direction, but it appears to be finite at the horizon!
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32. Example 20.8
Of course, this is in the coordinate frame and we need to project this vector onto the observers orthonormal basis to determine how much acceleration they feel. However, we can simply calculate the magnitude of the vector which is
Hence, the acceleration diverges as we try and hover closer and closer to the horizon (which is what we expect).
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