Aichelburg-Sexl ultraboost
Encyclopedia
In general relativity
, the Aichelburg–Sexl ultraboost is an exact solution
which models the physical experience of an observer moving past 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 metric tensor
can be written, in terms of Brinkmann coordinates
, as
The ultraboost can be obtained as the limit of various sequences of smooth Lorentzian manifolds.
For example, we can take Poor-man's Gaussian pulses
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
. As 
, 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.)
This resolves the following paradox: The moving particle will "think" that the stationary object (lets 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?
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, the Aichelburg–Sexl ultraboost is an exact solution
Exact solutions in general relativity
In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field....
which models the physical experience of an observer moving past a spherically symmetric gravitating object
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...
at nearly the speed of light. It was introduced by Peter C. Aichelburg
Peter C. Aichelburg
Peter C. Aichelburg is an Austrian physicist well-known for his contributions to general relativity, particularly the Aichelburg-Sexl ultraboost of the Schwarzschild vacuum....
and Roman U. Sexl in 1971.
The metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
can be written, in terms of Brinkmann coordinates
Brinkmann coordinates
Brinkmann coordinates are a particular coordinate system for a spacetime belonging to the family of pp-wave metrics. In terms of these coordinates, the metric tensor can be written asds^2 \, = H du^2 + 2 du dv + dx^2 + dy^2...
, as
The ultraboost can be obtained as the limit of various sequences of smooth Lorentzian manifolds.
For example, we can take Poor-man's Gaussian pulses
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
. As , 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.)This resolves the following paradox: The moving particle will "think" that the stationary object (lets 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?