[Paper Review] PP-WAVES (the original article)

Asher Peres|ArXiv.org|May 6, 2002

Black Holes and Theoretical Physics3 citations

TL;DR

This paper introduces a new class of exact solutions to Einstein's vacuum field equations representing gravitational waves with reduced symmetry, generalizing prior plane and cylindrical wave solutions. By constructing a metric with a harmonic function f(x,y,z+t), the authors derive wave solutions that belong to Petrov's second class, including a monochromatic plane wave and a localized wave packet, demonstrating the existence of regular, finite-duration gravitational wave packets with non-trivial spatial structure.

ABSTRACT

PP-waves have recently been of interest to string theorists. This is the original, hard to find, original article on plane polarized gravitational waves.

Motivation & Objective

• To derive a new class of exact solutions to Einstein's equations that generalize existing plane and cylindrical gravitational wave solutions by reducing symmetry.
• To explore the physical properties of gravitational waves with non-trivial spatial dependence in the transverse plane and time evolution.
• To investigate whether regular, finite-duration gravitational wave packets can exist in vacuum Einstein gravity.
• To classify the solutions within Petrov's curvature tensor classification scheme and analyze their physical components.

Proposed method

• The metric is defined as $ds^2 = -dx^2 - dy^2 - dz^2 + dt^2 - 2f(x,y,z+t)(dz + dt)^2$, with f harmonic in x and y.
• The Ricci tensor components are computed, showing that $R_{zz} = R_{zt} = R_{tt} = f_{xx} + f_{yy}$, vanishing when f is harmonic.
• An orthonormal tetrad is constructed to extract physical components of the Riemann curvature tensor, with $\tan(2a) = f_{xy}/f_{xx}$.
• The physical curvature components are derived as $\sigma = (f_{xx}^2 + f_{xy}^2)^{1/2}$, which are non-zero only for components involving the (3,4) directions.
• Two explicit examples are constructed: a monochromatic plane wave with $f = (x^2 - y^2)\sin(z+t)$, and a localized wave packet with $f = xy(x^2 + y^2)^{-2}\exp[-(b^2 - (z+t)^2)^{-2}]$ for $|z+t| < b$.
• The wave packet is shown to be smooth everywhere, including at the boundaries $|z+t| = b$, due to infinite differentiability of f.

Experimental results

Research questions

• RQ1 Can exact, regular gravitational wave solutions exist that are not plane or cylindrically symmetric?
• RQ2 What is the structure and behavior of gravitational waves with localized spatial and temporal profiles in vacuum Einstein gravity?
• RQ3 Do non-singular, finite-duration gravitational wave packets exist that satisfy the vacuum Einstein equations?
• RQ4 How do the curvature components and Petrov classification of such waves differ from known plane or cylindrical wave solutions?

Key findings

• The metric with $f = (x^2 - y^2)\sin(z+t)$ yields a monochromatic plane gravitational wave with amplitude $\sigma = 2\sin(z+t)$.
• The wave packet solution with $f = xy(x^2 + y^2)^{-2}\exp[-(b^2 - (z+t)^2)^{-2}]$ for $|z+t| < b$ is smooth and of compact support in time, traveling at unit velocity in the negative z-direction.
• The wave packet contains a singular segment along $x = y = 0$, $-b - t < z < b - t$, indicating a region of infinite curvature, though f is $C^\infty$ everywhere.
• The solution belongs to Petrov's second class, with only two independent non-vanishing physical curvature components $\sigma = (f_{xx}^2 + f_{xy}^2)^{1/2}$.
• When $f_{xx} + f_{yy} \neq 0$, the trace conditions $g^{mn}R_{mn} = 0$ and $g^{mn}R_{mr}R_{ns} = 0$ still hold, suggesting a possible interpretation as a source-free null electromagnetic field.