Gravitational properties of light: The emission of counter-propagating laser pulses from an atom

Dennis Rätzel, Martin Wilkens, and Ralf Menzel

Phys. Rev. D 95, 084008 – Published 6 April 2017

Abstract

The gravitational field of a laser pulse, although not detectable at the moment, has a special feature which continues to attract attention; cause and effect propagate with the same speed, the speed of light. One particular result of this feature is that the gravitational field of an emitted laser pulse and the gravitational effect of the emitter’s energy-momentum change are intimately entangled. In this article, a specific example of an emission process is considered: An atom, modeled as a point mass, emits two counter-propagating pulses. The corresponding curvature and the effect on massive and massless test particles is discussed. A comparison is made with the metric corresponding to a spherically symmetric massive object that isotropically emits radiation; the Vaidya metric.

Received 31 January 2017

DOI:https://doi.org/10.1103/PhysRevD.95.084008

Physics Subject Headings (PhySH)

General relativity Gravitation Lasers

Gravitation, Cosmology & AstrophysicsParticles & FieldsAtomic, Molecular & Optical

Authors & Affiliations

Dennis Rätzel, Martin Wilkens, and Ralf Menzel

University of Potsdam, Institute for Physics and Astronomy Karl-Liebknecht-Strasse 24/25, 14476 Potsdam, Germany

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Issue

Vol. 95, Iss. 8 — 15 April 2017

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Images

Figure 1
The laser pulses are modeled as pulses of electromagnetic radiation of length $L$ , traveling from the emitter along the $z$ -axis in the negative and positive $z$ -direction. The extension of the pulses in the transverse $x / y$ -directions is assumed to be negligible in comparison to their length.
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Figure 2
The pulses define world sheets restricted to the $t − z$ -plane. The world sheets touch along the world line of the emitter between the points A and B which correspond, respectively, to the start of the pulse emission and the end of the emission. The future directed light cones of A and B define the spacetime regions $I − I I I$ with qualitatively different metric perturbations. Region $I$ is the pre-emission cone, region $I I$ is the emission cone and region $I I I$ is the post-emission cone.
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Figure 3
The plots show one component of the metric perturbation $h_{00}$ for a point particle of mass $m c^{2} = 3 ε$ that emits two laser pulses of energy $ε$ and length $L$ in the coordinates $(c t, x, y, z)$ in the $(x, y)$ -plane for different times $t$ . $h_{00}$ is normalized to units of $κ$ and then $\log (1 + \log (1 + h_{00} / κ))$ is plotted. The $m c^{2}$ of the emitter is chosen close to the energy of the photons $ε$ such that the change in the gravitational field of the emitter due to the emission becomes visible. The description allows arbitrary masses as long as they are small enough to allow for the use of linearized gravity. In (a), we see the metric spherical symmetrical perturbation due to the point mass at rest at $(z, x) = (0, 0)$ . In (b), we see the effect of the two light pulses on the metric expanding from the point of their emission at $(z, x) = (0, 0)$ after the emission.
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Figure 4
The plots show the curvature components $R_{0 x 0 x}$ , $R_{0 x z x}$ , $R_{0 z 0 z}$ and $R_{z x z x}$ for the metric perturbation $h_{μ ν}$ induced by a point particle of mass $m c^{2} = 3 ε$ that emits two laser pulses of energy $ε$ in the coordinates $(c t, x, y, z)$ in the $(x, z)$ -plane for different times $t$ . The logarithm of value of the curvature components is encoded in the opacity of the color. Red is a negative value of curvature component and blue a positive value. White stands for zero.
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Figure 5
The Vaidya metric can be interpreted as a Schwarzschild metric with changing mass. The dependence of the mass term on the retarded time leads to a globally nonzero energy momentum tensor, which can be interpreted as a radially outward directed flux of electromagnetic radiation moving through the Schwarzschild background (represented by radial arrows in the figure).
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