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Problem 3: Show that the linearized Riemann tensor is invar...

Problem 3: Show that the linearized Riemann tensor is invariant under the action of a gravitational gauge transformation of the linearized metric variable hpp. Problem 4: Show that for a perfect fluid at rest in a stationary spherically symmetric gravitational field Spw with energy-momentum tens...

Problem 1: Consider a nonlinear coordinate transformation o...

Problem 1: Consider a nonlinear coordinate transformation of the form where (Taylo are the values of Lhe Christollel symbols al the origin It" = 0. Using the transformation law of the symbols Uxt show that under this transformation the values of the new Christoffel symbols 1"...

1 Problem I Identify a Killing vector of the metric ds&sup...

1 Problem I Identify a Killing vector of the metric ds² = −X² dT² + dX² Is this vector timelike, null or spacelike, and why? You should find that it is timelike away from X = 0, but null at X = 0. A horizon can be defined as a surface in spacetime at which a timelike Kil...

Problem I In static coordinates the de Sitter metric is 'd...

Problem I In static coordinates the de Sitter metric is 'dds)) where R is a constant. By analogy to the Eddingtor-Finkelstein coordinates of Schwarzschild space- time, let us set , = u + F(r), where the derivative F(r) is determined by the condition that grr = 0 in the resulting metric. Plea...

Problem I Consider a particle moving at fixed Rindler coord...

Problem I Consider a particle moving at fixed Rindler coordinate x = 1/g. a) What is the infinitesimal proper time dr in terms of dT? b) Compute the Minkowski components of the 4-velocity u° = dx°/dr by differentiating / = x sinh T. . = x coesh T. with respect to T and then multiplyin...

Problem Statement: For an antisymmetric tensor Fuv show that...

Problem Statement: For an antisymmetric tensor Fuv show that the combination Auvo = Fuv.o + + Fou.v transforms like a tensor of type (0,3) under a general curvilinear coordinate transformation x'v = x'v(xH). Comments: Since Fuv is antisymmetric, Fuv=-Fvu - (i) Remember, Fu've = axth 8x14 a...

Problem 4: Show that for a perfect fluid at rest in a stati...

Problem 4: Show that for a perfect fluid at rest in a stationary spherically symmetric grav- itational field guv with energy momentum tensor where ut" is the fluid 4-velocity, the covariant conservation law THE ; = 0 can be written as P.j. = - (p

First Problem Set Following our standard conventions for dr...

First Problem Set Following our standard conventions for drawing spacetime diagrams, draw four diagrams, with all the named points, hyperplanes, and individuals labeled on all four diagrams. Use graph paper. The first two diagrams represent a situation in Galilean spacetime, described below. Foll...

An excited iron (Fe) nucleus (mass 57 u) decays to it's grou...

An excited iron (Fe) nucleus (mass 57 u) decays to it's ground state with the emission of a photon. The energy available from this transition is 14.4KeV. A) By how much is the photon energy reduced from the full 14.4 KeV as a result of having to shareenergy with the recoiling atom? B) What is...

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