Gravitational Field of a
Large Moving Sheet

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Problem: Consider an infinitely large sheet of matter confined to the z = 0 plane. Assume the surface mass density, s_0, as measured in the zero momentum frame S₀ is nearly Newtonian, i.e. the components of stress-energy momentum tensor, T_mn as measured in the rest frame, satisfy the relations T₀₀ >> |T_0k| and T₀₀ >> |T_jk|. Assume also that the matter distribution in S₀ does not depend on time. Consider only the region of space above the sheet, i.e. only z > 0. In frame S, which is moving in the +x direction with speed v, the sheet is moving in the -x direction with speed v.

(a) Find the gravitational field, g_mn in S by transforming from S₀ to S using a background Lorentz transformation.

(b) Find dp_z /dt for u = 0 and z = 0 where u_z º dz/dt for a particle in free-fall.

(c) Find dp_z /dt for u_y = u_z = 0 and z = 0 for a particle in free-fall.

Solution (a): It can be shown that the gravitational field in S₀ is given by

Since the field is weak we employ a background Lorentz transformation, which is a transformation of the form

where

The transformation on the left is a boost in the +x direction with speed v. The transformation on the right is the inverse transformation. Transform the gravitational field using Eq. (2). The non-vanishing components are h₀₀, h₀₁, h₁₁, h₂₂, h₃₃. The components of the field in the new frame

Summary:

The metric in S then becomes

Solution (b): Acceleration in the weak field limit is given by

At z = 0, for u = 0 dt/dt = 1. U³ = -u_z and U⁰ = c. Eq. (37) then becomes

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