Aichelburg-Sexl ultraboost

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In general relativity, the Aichelburg/Sexl ultraboost is an exact solution which models the physical experience of an observer who whizzes by a spherically symmetric gravitating object at nearly the speed of light. It was introduced by Peter C. Aichelburg and Rudolph Sexl in 1971.

The metric tensor can be written, in terms of Brinkmann coordinates, as

<math> ds^2 = -8m \, \delta(u) \, \log r \, du^2 + 2 \, du \, dv + dr^2 + r^2 \, d\theta^2,</math>

<math> -\infty < u,r < 0, -\infty < v < \infty, \pi < \theta < \pi </math>

The ultraboost can be obtained as the limit of various sequences smooth Lorentzian manifolds. For example, we can take Poor-man's Gaussian pulses

<math> ds^2 = -\frac{4 m a \, \log(r)}{\pi \, (1+a^2 u^2)} \, du^2

- 2 du \, dv + dr^2 + r^2 \, d\theta^2 </math>

In these plus-polarized axisymmetric vacuum pp-waves, the curvature is concentrated along the axis of symmetry, falling off like O(m/r), and also near <math>u=0</math>. As <math>a \rightarrow \infty</math>, the wave profile turns into a Dirac delta, and we recover the ultraboost.

References

• {{{Author|}}}{{|{{{3}}}}}}|show1| (1998)}}{{{{{Year|}}}}}}|show1|.}} {{|{{{3}}}}}}|show1|[{{{URL}}}}} Black Hole Physics{{|{{{3}}}}}}|show1|]}}{{|{{{3}}}}}}|show1|, {{{Pages}}}}}{{|{{{3}}}}}}|Show1|, Boston: Klüwer}}. {{{ID|}}} See Section 7.6.12

• Poldolský, J.; & Griffiths, J. B. (1998). Boosted static multipole particles as sources of impulsive gravitational waves. Phys. Rev. D 58: 124024. See also "Boosted static multipole particles as sources of impulsive gravitational waves". ArXiv. URL accessed on June 15, 2005.

• Aichelburg, P. C.; & Sexl, R. U. (1971). On the gravitational field of a massless particle. Gen. Rel. Grav. 2: 303.