Gravitation Flashcards
What does it mean if a line element is proportional to the minkowski line element?
Angles are preserved
*another word for this I can’t remember
How can we show that an object is a tensor?
Show that it transforms like a tensor (i.e: with an appropriate jacobian matrix for each index)
*inverse jacobian matrix for lowered indices
How does a metric tensor transform?
What is the definition of the jacobian matrix?
Check wall notes (not written out explicitly).
Also make sure to get the definition of the jacobian matrix.
What is the difference between an ordinary derivative and a covariant derivative?
What about an absolute derivative (along a curve)?
Mathematical definition is given in the formula sheet.
A covariant derivative tells us about the genuine change in a function over a manifold, including effects from the changing basis vectors with a term in the affine connection.
Absolute derivative parametrises change along a curve - then use chain rule with a covariant derivative.
What does the affine connection describe?
Describes how basis vectors vary infinitesimally over the manifold.
*symmetric in bottom two indices
*not a tensor (doesn’t transform like a tensor)
What is the equation of an affinely-parametrised affine geodesic?
How can it be derived?
Check wall notes
Derived via transforming N2: a = 0 into curved space from flat space.
How can we find the connection by comparing formulae for affine and metric connections.
Apply E-L equations to the effective lagrangian.
L_eff given in Wall notes.
*this gives the minima of the line element between two points: METRIC connection.
Compare this to the equation of an affinely-parametrised affine geodesic to deduce the components of the connection.
*In a tortion-free manifold, metric and affine connections are identical.
How can we lower / raise indices?
Apply the metric / inverse metric.
What is the connection for minkowski (flat) space?
0
What is the significance of the effective Lagrangian where the geodesic is parametrised by proper time?
Constant along the geodesic, equal to either:
+1 : time-like geodesic (trajectory of massive particle)
0 : light-like geodesic (trajectory of light)
-1 : space-like geodesic (unphysical)
L_eff can be shown to be constant by differentiating w.r.t. the geodesic parameter, and applying the equation of an affinely-parametrised affine geodesic.
*also must use the form of the christoffel symbols
What is the Ricci identity?
(What does the Riemann Tensor represent?)
The commutator of covariant derivatives acting on a vector is equal to the riemann tensor acting on the vector [check wall notes].
The riemann tensor represents the difference between parallel transporting along different paths to the same destination.
What is the relationship between the Ricci tensor and the Riemann tensor?
What about the Ricci scalar?
The Ricci tensor (2-tensor) is the only non-zero contraction of the Riemann tensor (1st and 3rd indices).
In 2D and 3D the Ricci tensor completely describes the Riemann tensor.
The Ricci scalar is the only non-zero contraction of the Ricci tensor (trace). Referred to sometimes as the curvature.
In 2D the Ricci scalar completely describes the Riemann tensor.
What are the symmetries of the Riemann tensor?
Considering Riemann tensor with all indices lowered:
Antisymmetric in 1 <=> 2 or 3 <=> 4.
Symmetric in 1,2 <=> 3,4.
Cyclic symmetry in 2,3,4 (sum of all three cycles = 0)
What is the definition of the Einstein tensor and what are its key properties?
Check wall notes!
Ricci tensor - 1/2 * metric tensor * Ricci scalar
The Einstein tensor summarises the curvature of a manifold in a covariantly-conserved way. (i.e: covariant derivative of the tensor = 0)
What is a general expression for the stress-energy tensor?
-for a dust
-for a fluid
Dust: co-moving energy density / c^2 * two 4-velocities with different indices.
Fluid: Same as dust but density + pressure instead of density. Also extra term: - Pressure * metric tensor
How do we know that the stress-energy tensor is covariantly conserved?
Covariant conservation of the stress-energy tensor is equivalent to the continuity equation (energy conservation).
+ energy flux <=> momentum density
momentum density = -d/dx stress tensor
What is Einstein’s equation?
How do we get to the trace-reversed version?
Einstein tensor = K * Stress-energy tensor
Contract with inverse metric to get to trace-reversed version.
K = 8 pi G_N / c^4 recovers Newtonian gravity in the non-relativistic limit.
How can we parametrise the weak gravity limit?
Metric equal to minkowski metric + h
where h is a metric with determinant much less than 1
( - h for the inverse metric)
Expand in h
The connection can be written simply in terms of h.
What is the source term for gravitational waves?
The second time derivative of the moment of inertia (second moment of mass).
*i.e: fast acceleration with large oscillation in quadrupole moment produces large amplitude gravitational waves.
What steps do we take to reach the schwarzschild solution?
Use general spherically-symmetric line element.
ds^2 = e^nu dt^2 - e^lambda dr^2 - r^2(dtheta^2 + sin^2(theta)dphi^2)
* nu and lambda dependent on r and t generally.
Find the connection using L_eff.
Construct the Riemann tensor -> Ricci tensor and scalar
Apply the free space Einstein’s equation.
Use gauge freedom to remove final free parameter (change t coordinate).
What is the schwarszchild radius?
2GM / c^2
What is Birkhoff’s theorem?
Spherically symmetric vacuum solutions can always be made static by a suitable choice of variables.
* so spherically symmetric mass distributions cannot emit gravitational waves.
What are the 4 classical tests of GR in the solar system?
Gravitational redshift
Light deflection
Perihelion precession
Light delay
At what rate does an observer at fixed coordinates in a gravitational well measure time passing?
How is this related to the frequency of light they emit/detect?
Consider schwarzschild line element for fixed spatial coordinates.
In this case the line element is equal to proper time.
dt / dtau gives the rate at which time is measured to pass. This is directly proportional to the frequency of light that is measured/emitted.
Consider the ratio of these rates for different positions in the well. For a distant observer, consider one of the positions as infinitely far from the source.
Geodesics for schwarzschild solution (given in formula sheet).
-what does l refer to?
-which is the non-classical term?
-which term should be considered the effective potential?
l is the angular momentum (per unit mass):
r^2 phi(dot)
The non-classical part is the term in l^2 r_s / r^2
The term in square brackets is the effective potential. The other term on the RHS is a constant - total energy.
What is the requirement for a stable circular orbit?
Energy = Potential at that radius.
Potential’ = 0
Potential’’ > 0 (otherwise unstable)
i.e: the bottom of a trough in V_eff
What property do particles coming from infinity have? (orbital dynamics)
E > 0
How can we consider the orbital dynamics of a photon?
Set K = 0 (null geodesic (light-like))
Set epsilon = 1 {it is really more complicated than this, we must reparametrise as tau is no longer a good choice of coordinate (->inf for massless). This reparametrisation is equivalent to setting epsilon = 1}
