Lemaître coordinates are a particular set of coordinates for the
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
—a spherically symmetric solution to the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
in vacuum—introduced by
Georges Lemaître
Georges Henri Joseph Édouard Lemaître ( ; ; 17 July 1894 – 20 June 1966) was a Belgian Catholic priest, theoretical physicist, mathematician, astronomer, and professor of physics at the Catholic University of Louvain. He was the first to th ...
in 1932.
[ English translation: ]
See also: … Changing from
Schwarzschild to Lemaître coordinates removes the
coordinate singularity A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame that can be removed by choosing a different frame.
An example is the apparent (longitudinal) singularity at the 90 degree latitude in sph ...
at the
Schwarzschild radius
The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic ...
.
Equations
The original Schwarzschild coordinate expression of the Schwarzschild metric, in
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a Coherence (units of measurement), coherent unit of a quantity. For e ...
(), is given as
:
where
:
is the
invariant interval
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
;
:
is the Schwarzschild radius;
:
is the mass of the central body;
:
are the
Schwarzschild coordinates In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coord ...
(which asymptotically turn into the flat
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
);
:
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
;
:and
is the
gravitational constant
The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
.
This metric has a coordinate singularity at the Schwarzschild radius
.
Georges Lemaître was the first to show that this is not a real physical singularity but simply a manifestation of the fact that the static Schwarzschild coordinates cannot be realized with material bodies inside the Schwarzschild radius. Indeed, inside the Schwarzschild radius everything falls towards the centre and it is impossible for a physical body to keep a constant radius.
A transformation of the Schwarzschild coordinate system from
to the new coordinates
:
(the numerator and denominator are switched inside the square-roots), leads to the Lemaître coordinate expression of the metric,
:
where
:
The trajectories with ''ρ'' constant are
timelike geodesics with ''τ'' the proper time along these geodesics. They represent the motion of freely falling particles which start out with zero velocity at infinity. At any point their speed is just equal to the escape velocity from that point.
In Lemaître coordinates there is no singularity at the Schwarzschild radius, which instead corresponds to the point
. However, there remains a genuine
gravitational singularity
A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravitational field, gravity is so intense that spacetime itself breaks down catastrophically. As such, a singularity is by definition no longer p ...
at the center, where
, which cannot be removed by a coordinate change.
The Lemaître coordinate system is
synchronous
Synchronization is the coordination of events to operate a system in unison. For example, the conductor of an orchestra keeps the orchestra synchronized or ''in time''. Systems that operate with all parts in synchrony are said to be synchronou ...
, that is, the global time coordinate of the metric defines the proper time of co-moving observers. The radially falling bodies reach the Schwarzschild radius and the centre within finite proper time.
Along the trajectory of a radial light ray,
:
therefore no signal can escape from inside the Schwarzschild radius, where always
and the light rays emitted radially inwards and outwards both
end up at the origin.
See also
*
Kruskal-Szekeres coordinates
*
Eddington–Finkelstein coordinates In general relativity, Eddington–Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry (e.g. a spherically symmetric black hole) which are adapted to radial null geodesics. Null geodesics are the worldlines of p ...
*
Lemaître–Tolman metric
In physics, the Lemaître–Tolman metric, also known as the Lemaître–Tolman–Bondi metric or the Tolman metric, is a Lorentzian metric based on an exact solution of Einstein's field equations; it describes an isotropic and expanding (or co ...
*
Introduction to the mathematics of general relativity
The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solve ...
*
Stress–energy tensor
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
*
Metric tensor (general relativity)
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The me ...
*
Relativistic angular momentum
In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the thr ...
References
{{DEFAULTSORT:Lemaitre coordinates
Metric tensors
Spacetime
Coordinate charts in general relativity
General relativity
Gravity
Exact solutions in general relativity