View source for Aichelburg–Sexl ultraboost

{{No footnotes|date=November 2022}}
In [[general relativity]], the '''Aichelburg–Sexl ultraboost''' is an [[Exact solutions in general relativity|exact solution]] which models the spacetime of an  observer moving towards or away from a [[Schwarzschild metric|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]] 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,v < \infty, \, 0 < r < \infty, \, -\pi < \theta < \pi </math>
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.
:<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 <math>O(m/r)</math>, and also near <math>u=0</math>.  As <math>a \rightarrow \infty</math>, the wave profile turns into a [[Dirac delta]] 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 ==
* {{cite book | author=Frolov, Valeri P. | author2=  Novikov, Igor D. | name-list-style= amp | title=Black Hole Physics | publisher=Klüwer | location=Boston | date=1998 | isbn=0-7923-5146-0}}  ''See Section 7.6.12''
* {{cite journal | author=Podolský, J. | author2= Griffiths, J. B. | name-list-style= amp | title=Boosted static multipole particles as sources of impulsive gravitational waves | journal=Phys. Rev. D | date=1998 | volume=58 | issue= 12 | pages=124024 | doi=10.1103/PhysRevD.58.124024 | arxiv=gr-qc/9809003 | bibcode=1998PhRvD..58l4024P | s2cid= 48232013 | url=https://core.ac.uk/download/pdf/2737996.pdf}}
* {{cite journal | author=Aichelburg, P. C. | authorlink1=Peter C. Aichelburg | author2=Sexl, R. U. |authorlink2= Roman U. Sexl | name-list-style=amp | title=On the gravitational field of a massless particle | journal=General Relativity and Gravitation | date=December 1971 | volume=2 | issue=4 | pages=303–312 | doi=10.1007/BF00758149|bibcode = 1971GReGr...2..303A | s2cid=123066007 }}

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[[Category:Exact solutions in general relativity]]