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general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, the Aichelburg–Sexl ultraboost is an exact solution which models the spacetime of an observer moving towards or away from a spherically symmetric gravitating object at nearly the speed of light. It was introduced by Peter C. Aichelburg and Roman U. Sexl in 1971. The original motivation behind the ultraboost was to consider the gravitational field of massless point particles within general relativity. It can be considered an approximation to the gravity well of a photon or other lightspeed particle, although it does not take into account quantum uncertainty in particle position or momentum. The
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
can be written, in terms of Brinkmann coordinates, as : ds^2 = -8m \, \delta(u) \, \log r \, du^2 + 2 \, du \, dv + dr^2 + r^2 \, d\theta^2, : -\infty < u,v < \infty, \, 0 < r < \infty, \, -\pi < \theta < \pi The ultraboost can be obtained as the limit of a metric, which is also an exact solution, at least if one admits impulsive curvatures. For example, one can take a Gaussian pulse. : ds^2 = -\frac \, du^2 + 2 du \, dv + dr^2 + r^2 \, d\theta^2, 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 u=0. As a \rightarrow \infty, the wave profile turns into a
Dirac delta In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
and the ultraboost is recovered. The ultraboost helps also to understand why fast moving observers won't see moving stars and planet-like objects become black holes.


References

* ''See Section 7.6.12'' * * {{DEFAULTSORT:Aichelburg-Sexl Ultraboost Exact solutions in general relativity