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| In [[general relativity]], the '''Aichelburg–Sexl ultraboost''' is an [[Exact solutions in general relativity|exact solution]] which models the physical experience of an observer moving past 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.
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| The [[metric tensor]] can be written, in terms of [[Brinkmann coordinates]], as
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| :<math> ds^2 = -8m \, \delta(u) \, \log r \, du^2 + 2 \, du \, dv + dr^2 + r^2 \, d\theta^2,</math>
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| :<math> -\infty < u < \infty, \, 0 < r < \infty, \, -\infty < v < \infty, -\pi < \theta < \pi </math>
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| The ultraboost can be obtained as the limit of various sequences of smooth Lorentzian manifolds.
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| For example, we can take ''Poor-man's Gaussian pulses''
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| :<math> ds^2 = -\frac{4 m a \, \log(r)}{\pi \, (1+a^2 u^2)} \, du^2
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| - 2 du \, dv + dr^2 + r^2 \, d\theta^2, </math>
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| :<math> -\infty < u < \infty, \, 0 < r < \infty, \, -\infty < v < \infty, -\pi < \theta < \pi </math>
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| 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. (To avoid possible misunderstanding, we stress that these are exact solutions which approximate the ultraboost, which is also an exact solution, at least if you admit impulsive curvatures.)
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| This resolves the following paradox: The moving particle will "think" that the stationary object (let's use a planet) has a huge mass, because in the particle's point of view the planet is moving at an ultra relativistic speed. What if the particle moves fast enough so that the planet becomes a black hole, and the particle gets inside the event horizon? Why does it fly right past (like a photon) and not get trapped?
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| == References ==
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| * {{cite book | author=Frolov, Valeri P.; & Novikov, Igor D. | title=Black Hole Physics | publisher=Klüwer | location=Boston | year=1998 | isbn=0-7923-5146-0}} ''See Section 7.6.12''
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| * {{cite journal | author=Poldolský, J.; & Griffiths, J. B. | title=Boosted static multipole particles as sources of impulsive gravitational waves | journal=Phys. Rev. D | year=1998 | volume=58 | pages=124024 | doi=10.1103/PhysRevD.58.124024 |arxiv=gr-qc/9809003|bibcode = 1998PhRvD..58l4024P }}
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| * {{cite journal | author=Aichelburg, P. C.; & Sexl, R. U. | title=On the gravitational field of a massless particle | journal=Gen. Rel. Grav. | year=1971 | volume=2 | pages=303 | doi=10.1007/BF00758149|bibcode = 1971GReGr...2..303A }}
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| {{DEFAULTSORT:Aichelburg-Sexl Ultraboost}}
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| [[Category:Exact solutions in general relativity]]
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