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PHY 600 Prob Set 6: Weak-Field Metric, Parallel Transport, Spherical Metrics - Prof. Alfre, Assignments of Physics

University of Kentucky (UK)Physics
Prof. Alfred D. Shapere

Problem set 6 for phy 600, due on 6 april 2009. The problems involve calculating the weak-field metric, understanding parallel transport on a sphere, and finding the most general spherically symmetric vacuum solution to the einstein equations. Students are expected to use mathematica or maple for some problems.

Typology: Assignments

Pre 2010

Uploaded on 10/01/2009

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koofers-user-zic 🇺🇸

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PHY 600
Problem Set #6
due 6 April 2009
1. Hartle 8-12.
2. Hartle 21-11. Do this using Mathematica or Maple.
3. The weak-field metric: Show that the weak-field metric eq.(21.25) is the unique
solution to the equations of linearized gravity, which is time-independent,
diagonal, and approaches
η
αβ
far from any sources. Proceed by solving (21.55)
and (21.56) for h
αβ
satisfying these conditions.
4. On a two-dimensional sphere of radius a, a vector is transported around a
triangular path whose edges are geodesics.
(a) Derive the relation between the rotation of the vector under parallel
transport and the area of the triangle. Generalize to a path of arbitrary
shape.
(b) Compute the components of the Riemann curvature tensor on the sphere
by hand.
(c) Show that your result in (b) reproduces the relation found in (a), for paths
enclosing a small area.
5. Themostgeneralstatic,sphericallysymmetricmetricin4dimensionscanbe
writtenintheform
ds2=–f(r)dt2+g(r)dr2+r2(d
θ
2+sin2
θ
d
φ
2)
ComputethenecessaryChristoffelsymbolsandcurvaturecomponentsby
handandfindthemostgeneralsphericallysymmetricvacuumsolutiontothe
acuumEinstein’sequations(awayfromr=0).Followtheargumentin
xample21.4.
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Download PHY 600 Prob Set 6: Weak-Field Metric, Parallel Transport, Spherical Metrics - Prof. Alfre and more Assignments Physics in PDF only on Docsity!

PHY 600

Problem Set

due 6 April 2009

  1. Hartle 8-12.
  2. Hartle 21-11. Do this using Mathematica or Maple.
  3. The weak-field metric : Show that the weak-field metric eq.(21.25) is the unique solution to the equations of linearized gravity, which is time-independent,

diagonal, and approaches η αβ far from any sources. Proceed by solving (21.55)

and (21.56) for h αβ satisfying these conditions.

  1. On a two-dimensional sphere of radius a , a vector is transported around a triangular path whose edges are geodesics.

(a) Derive the relation between the rotation of the vector under parallel transport and the area of the triangle. Generalize to a path of arbitrary shape.

(b) Compute the components of the Riemann curvature tensor on the sphere by hand.

(c) Show that your result in (b) reproduces the relation found in (a), for paths enclosing a small area.

  1. The most general static, spherically symmetric metric in 4 dimensions can be written in the form

ds^2 = – f ( r ) dt^2 + g ( r ) dr^2 + r^2 ( d θ 2 + sin 2 θ d φ 2 )

Compute the necessary Christoffel symbols and curvature components by hand and find the most general spherically symmetric vacuum solution to the acuum Einstein’s equations (away from r = 0). Follow the argument in xample 21.4.

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