general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...

and gravitation the Palatini variation is nowadays thought of as a variation of a

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

with respect to the connection. In fact, as is well known, the

Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the act ...

for general relativity was first formulated purely in terms of the

spacetime metric 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 ...

. In the Palatini variational method one takes as independent field variables not only the ten components

but also the forty components of the

affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values ...

, assuming, a priori, no dependence of the

from the

and their derivatives. The reason the Palatini variation is considered important is that it means that the use of the

Christoffel connection In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distan ...

in general relativity does not have to be added as a separate assumption; the information is already in the Lagrangian. For theories of gravitation which have more complex Lagrangians than the Einstein–Hilbert Lagrangian of general relativity, the Palatini variation sometimes gives more complex connections and sometimes tensorial equations.

Attilio Palatini Attilio Palatini (18 November 1889 – 24 August 1949) was an Italian mathematician born in Treviso. Biography Palatini was the seventh of the eight children of Michele (1855-1914) and Ilde Furlanetto (1856-1895). In 1900, during the celebrations ...

(1889–1949) was an Italian mathematician who received his doctorate from the

University of Padova The University of Padua ( it, Università degli Studi di Padova, UNIPD) is an Italian university located in the city of Padua, region of Veneto, northern Italy. The University of Padua was founded in 1222 by a group of students and teachers fro ...

, where he studied under

Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significa ...

and Ricci-Curbastro. The history of the subject, and Palatini's connection with it, are not straightforward (see references). In fact, it seems that what the textbooks now call "

Palatini formalism Palatini may refer to: * Attilio Palatini (1889–1949), Italian mathematician * (1855–1914), Italian politician * Palatini identity * Palatini variation * Latin plural of Palatine A palatine or palatinus (in Latin; plural ''palatini''; cf ...

" was actually invented in 1925 by Einstein, and as the years passed, people tended to mix up the

Palatini identity In general relativity and tensor calculus, the Palatini identity is: : \delta R_ = \nabla_\rho (\delta \Gamma^\rho_) - \nabla_\nu (\delta \Gamma^\rho_), where \delta \Gamma^\rho_ denotes the variation of Christoffel symbols and \nabla_\rho indi ...

and the Palatini formalism.

References

* P. G. Bergmann and V. De Sabbata (eds.) Cosmology and Gravitation, Plenum Press, New York (1980)">Peter_Bergmann.html" ;"title="nglish translation by R. Hojman and C. Mukku in Peter Bergmann">P. G. Bergmann and V. De Sabbata (eds.) Cosmology and Gravitation, Plenum Press, New York (1980)* * Lagrangian mechanics General relativity {{relativity-stub

See also

References