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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. ...
, a geodesic generalizes the notion of a "straight line" to curved
spacetime 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 diffe ...
. Importantly, the
world line The world line (or worldline) of an object is the path that an object traces in 4- dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from c ...
of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic. In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress–energy tensor (representing matter, for instance). Thus, for example, the path of a planet orbiting a star is the projection of a geodesic of the curved four-dimensional (4-D) spacetime geometry around the star onto three-dimensional (3-D) space.


Mathematical expression

The full geodesic equation is : +\Gamma^\mu _=0\ where ''s'' is a scalar parameter of motion (e.g. the proper time), and \Gamma^\mu _ are Christoffel symbols (sometimes called the affine connection coefficients or
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
coefficients) symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and the summation convention is used for repeated indices \alpha and \beta. The quantity on the left-hand-side of this equation is the acceleration of a particle, so this equation is analogous to
Newton's laws of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
, which likewise provide formulae for the acceleration of a particle. The Christoffel symbols are functions of the four spacetime coordinates and so are independent of the velocity or acceleration or other characteristics of a test particle whose motion is described by the geodesic equation.


Equivalent mathematical expression using coordinate time as parameter

So far the geodesic equation of motion has been written in terms of a scalar parameter ''s''. It can alternatively be written in terms of the time coordinate, t \equiv x^0 (here we have used the triple bar to signify a definition). The geodesic equation of motion then becomes: : =- \Gamma^\mu _+ \Gamma^0 _\ . This formulation of the geodesic equation of motion can be useful for computer calculations and to compare General Relativity with Newtonian Gravity. It is straightforward to derive this form of the geodesic equation of motion from the form which uses proper time as a parameter using the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
. Notice that both sides of this last equation vanish when the mu index is set to zero. If the particle's velocity is small enough, then the geodesic equation reduces to this: : =- \Gamma^n _. Here the Latin index ''n'' takes the values ,2,3 This equation simply means that all test particles at a particular place and time will have the same acceleration, which is a well-known feature of Newtonian gravity. For example, everything floating around in the
International Space Station The International Space Station (ISS) is the largest Modular design, modular space station currently in low Earth orbit. It is a multinational collaborative project involving five participating space agencies: NASA (United States), Roscosmos ( ...
will undergo roughly the same acceleration due to gravity.


Derivation directly from the equivalence principle

Physicist Steven Weinberg has presented a derivation of the geodesic equation of motion directly from the equivalence principle.Weinberg, Steven. ''Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity'' (Wiley 1972). The first step in such a derivation is to suppose that a free falling particle does not accelerate in the neighborhood of a point-event with respect to a freely falling coordinate system (X^\mu). Setting T \equiv X^0, we have the following equation that is locally applicable in free fall: : = 0 . The next step is to employ the multi-dimensional
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
. We have: : = Differentiating once more with respect to the time, we have: : = + We have already said that the left-hand-side of this last equation must vanish because of the Equivalence Principle. Therefore: : =- Multiply both sides of this last equation by the following quantity: : Consequently, we have this: : = - \left \right. Weinberg defines the affine connection as follows: :\Gamma^\lambda _ = \left \right/math> which leads to this formula: : = - \Gamma^_ . Notice that, if we had used the proper time “s” as the parameter of motion, instead of using the locally inertial time coordinate “T”, then our derivation of the geodesic equation of motion would be complete. In any event, let us continue by applying the one-dimensional
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
: : \left( \frac \right)^2 + \frac = - \Gamma^_ \left( \frac \right)^2 . : + \frac \left( \frac \right)^2 = - \Gamma^_ . As before, we can set t \equiv x^0. Then the first derivative of ''x''0 with respect to ''t'' is one and the second derivative is zero. Replacing ''λ'' with zero gives: : \frac \left( \frac \right)^2 = - \Gamma^_ . Subtracting d ''x''''λ'' / d ''t'' times this from the previous equation gives: : = - \Gamma^_ + \Gamma^_ which is a form of the geodesic equation of motion (using the coordinate time as parameter). The geodesic equation of motion can alternatively be derived using the concept of
parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
.


Deriving the geodesic equation via an action

We can (and this is the most common technique) derive the geodesic equation via the action principle. Consider the case of trying to find a geodesic between two timelike-separated events. Let the action be :S=\int ds where ds=\sqrt is the line element. There is a negative sign inside the square root because the curve must be timelike. To get the geodesic equation we must vary this action. To do this let us parameterize this action with respect to a parameter \lambda. Doing this we get: :S=\int\sqrt \, d\lambda We can now go ahead and vary this action with respect to the curve x^. By the principle of least action we get: :0=\delta S=\int\delta\left(\sqrt\right) \, d\lambda =\int\fracd\lambda Using the product rule we get: :0=\int\left(\frac\frac\delta g_+g_\frac\frac + g_ \frac \frac\right) \, d\lambda = \int\left(\frac\frac \partial_\alpha g_ \delta x^\alpha +2g_\frac\frac\right) \, d\lambda where : \frac = \sqrt Integrating by-parts the last term and dropping the total derivative (which equals to zero at the boundaries) we get that: :0=\int \left(\frac\frac\partial_\alpha g_\delta x^\alpha-2\delta x^\mu\frac \left(g_ \frac\right)\right) \, d\tau = \int \left(\frac\frac \partial_\alpha g_\delta x^\alpha-2\delta x^\mu \partial_\alpha g_\frac\frac-2\delta x^\mu g_\frac\right) \, d\tau Simplifying a bit we see that: :0=\int \left(-2g_\frac+\frac\frac \partial_\mu g_ - 2\frac \frac \partial_\alpha g_\right) \delta x^\mu d\tau so, :0=\int \left(-2g_\frac+\frac\frac\partial_\mu g_-\frac\frac\partial_\alpha g_-\frac \frac \partial_\nu g_\right) \delta x^\mu \, d\tau multiplying this equation by -\frac we get: :0=\int \left(g_\frac+\frac \frac\frac\left(\partial_\alpha g_ + \partial_\nu g_-\partial_\mu g_\right)\right) \delta x^\mu \, d\tau So by Hamilton's principle we find that the Euler–Lagrange equation is :g_\frac+\frac\frac\frac\left(\partial_g_+\partial_g_-\partial_g_\right)=0 Multiplying by the inverse metric tensor g^ we get that :\frac+\fracg^\left(\partial_g_+\partial_g_-\partial_g_\right)\frac\frac=0 Thus we get the geodesic equation: :\frac+\Gamma^_\frac\frac=0 with the Christoffel symbol defined in terms of the metric tensor as :\Gamma^_=\fracg^\left(\partial_g_+\partial_g_-\partial_g_\right) (Note: Similar derivations, with minor amendments, can be used to produce analogous results for geodesics between light-like or space-like separated pairs of points.)


Equation of motion may follow from the field equations for empty space

Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
believed that the geodesic equation of motion can be derived from the field equations for empty space, i.e. from the fact that the Ricci curvature vanishes. He wrote:
It has been shown that this law of motion — generalized to the case of arbitrarily large gravitating masses — can be derived from the field equations of empty space alone. According to this derivation the law of motion is implied by the condition that the field be singular nowhere outside its generating mass points.
and
One of the imperfections of the original relativistic theory of gravitation was that as a field theory it was not complete; it introduced the independent postulate that the law of motion of a particle is given by the equation of the geodesic. A complete field theory knows only fields and not the concepts of particle and motion. For these must not exist independently from the field but are to be treated as part of it. On the basis of the description of a particle without singularity, one has the possibility of a logically more satisfactory treatment of the combined problem: The problem of the field and that of the motion coincide.
Both physicists and philosophers have often repeated the assertion that the geodesic equation can be obtained from the field equations to describe the motion of a gravitational singularity, but this claim remains disputed. According to David Malament, “Though the geodesic principle can be recovered as theorem in general relativity, it is not a consequence of Einstein’s equation (or the conservation principle) alone. Other assumptions are needed to derive the theorems in question.” Less controversial is the notion that the field equations determine the motion of a fluid or dust, as distinguished from the motion of a point-singularity.


Extension to the case of a charged particle

In deriving the geodesic equation from the equivalence principle, it was assumed that particles in a local inertial coordinate system are not accelerating. However, in real life, the particles may be charged, and therefore may be accelerating locally in accordance with the
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an ele ...
. That is: : = . with : =-1. The
Minkowski tensor In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
\eta_ is given by: :\eta_ = \begin-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end These last three equations can be used as the starting point for the derivation of an equation of motion in General Relativity, instead of assuming that acceleration is zero in free fall. Because the Minkowski tensor is involved here, it becomes necessary to introduce something called the '' metric tensor'' in General Relativity. The metric tensor ''g'' is symmetric, and locally reduces to the Minkowski tensor in free fall. The resulting equation of motion is as follows: : =- \Gamma^\mu _\ + . with : =-1. This last equation signifies that the particle is moving along a timelike geodesic; massless particles like the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
instead follow null geodesics (replace −1 with zero on the right-hand side of the last equation). It is important that the last two equations are consistent with each other, when the latter is differentiated with respect to proper time, and the following formula for the Christoffel symbols ensures that consistency: :\Gamma^_=\fracg^ \left(\frac + \frac - \frac \right) This last equation does not involve the electromagnetic fields, and it is applicable even in the limit as the electromagnetic fields vanish. The letter ''g'' with superscripts refers to the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...
of the metric tensor. In General Relativity, indices of tensors are lowered and raised by contraction with the metric tensor or its inverse, respectively.


Geodesics as curves of stationary interval

A geodesic between two events can also be described as the curve joining those two events which has a stationary interval (4-dimensional "length"). ''Stationary'' here is used in the sense in which that term is used in the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, namely, that the interval along the curve varies minimally among curves that are nearby to the geodesic. In Minkowski space there is only one geodesic that connects any given pair of events, and for a time-like geodesic, this is the curve with the longest proper time between the two events. In curved spacetime, it is possible for a pair of widely separated events to have more than one time-like geodesic between them. In such instances, the proper times along several geodesics will not in general be the same. For some geodesics in such instances, it is possible for a curve that connects the two events and is nearby to the geodesic to have either a longer or a shorter proper time than the geodesic. For a space-like geodesic through two events, there are always nearby curves which go through the two events that have either a longer or a shorter proper length than the geodesic, even in Minkowski space. In Minkowski space, the geodesic will be a straight line. Any curve that differs from the geodesic purely spatially (''i.e.'' does not change the time coordinate) in any inertial frame of reference will have a longer proper length than the geodesic, but a curve that differs from the geodesic purely temporally (''i.e.'' does not change the space coordinates) in such a frame of reference will have a shorter proper length. The interval of a curve in spacetime is : l = \int \sqrt \, ds\ . Then, the Euler–Lagrange equation, : \sqrt = \sqrt \ , becomes, after some calculation, : 2\left(\Gamma^\lambda _ \dot x^\mu \dot x^\nu + \ddot x^\lambda\right) = U^\lambda \ln , U_\nu U^\nu, \ , where U^\mu = \dot x^\mu . The goal being to find a curve for which the value of : l = \int d\tau = \int \, d\phi = \int \sqrt \, d\phi = \int \sqrt \, d\phi = \int f \, d\phi is stationary, where : f = \sqrt such goal can be accomplished by calculating the Euler–Lagrange equation for ''f'', which is : = . Substituting the expression of ''f'' into the Euler–Lagrange equation (which makes the value of the integral ''l'' stationary), gives : = Now calculate the derivatives: \left( \right) = \qquad \qquad (1) \left( \right) = \qquad \qquad (2) \left( \right) = \qquad \qquad (3) = \qquad \qquad (4) = g_ \dot x^\mu \dot x^\nu \qquad \qquad (5) (g_ \dot x^\mu \dot x^\nu) (g_ \dot x^\nu \dot x^\mu + g_ \dot x^\mu \dot x^\nu + g_ \ddot x^\nu + g_ \ddot x^\mu) := (g_ \dot x^\mu \dot x^\nu) (g_ \dot x^\alpha \dot x^\beta) + (g_ \dot x^\nu + g_ \dot x^\mu) (g_ \dot x^\mu \dot x^\nu) \qquad \qquad (6) g_ \dot x^\mu \dot x^\nu + g_ \dot x^\mu \dot x^\nu - g_ \dot x^\mu \dot x^\nu + 2 g_ \ddot x^\mu = \qquad \qquad (7) 2(\Gamma_ \dot x^\mu \dot x^\nu + \ddot x_\lambda) = = = U_\lambda \ln , U_\nu U^\nu, \qquad \qquad (8) This is just one step away from the geodesic equation. If the parameter ''s'' is chosen to be affine, then the right side of the above equation vanishes (because U_\nu U^\nu is constant). Finally, we have the geodesic equation : \Gamma^\lambda _ \dot x^\mu \dot x^\nu + \ddot x^\lambda = 0\ .


Derivation using autoparallel transport

The geodesic equation can be alternatively derived from the autoparallel transport of curves. The derivation is based on the lectures given by Frederic P. Schuller at the We-Heraeus International Winter School on Gravity & Light. Let (M,O,A,\nabla) be a smooth manifold with connection and \gamma be a curve on the manifold. The curve is said to be autoparallely transported if and only if \nabla_v_=0 . In order to derive the geodesic equation, we have to choose a chart (U,x) \in A: : \nabla_ \left( \dot \gamma^m \frac \right)=0 Using the C^ linearity and the Leibniz rule: : \dot \gamma^i \left( \nabla_ \dot \gamma^m \right) \frac+\dot \gamma^i \dot \gamma^m \nabla_\left( \frac \right)=0 Using how the connection acts on functions (\dot \gamma^m ) and expanding the second term with the help of the connection coefficient functions: : \dot \gamma^i \frac \frac+\dot \gamma^i \dot \gamma^m \Gamma^_ \frac=0 The first term can be simplified to \ddot \gamma^m \frac . Renaming the dummy indices: : \ddot \gamma^q \frac+\dot \gamma^i \dot \gamma^m \Gamma^_ \frac=0 We finally arrive to the geodesic equation: : \ddot \gamma^q +\dot \gamma^i \dot \gamma^m \Gamma^_=0


See also

*
Geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
*
Geodetic precession The geodetic effect (also known as geodetic precession, de Sitter precession or de Sitter effect) represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body. For example, ...
* Schwarzschild geodesics * Geodesics as Hamiltonian flows * Synge's world function


Bibliography

* Steven Weinberg, ''Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity'', (1972) John Wiley & Sons, New York . ''See chapter 3''. *
Lev D. Landau Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet- Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of theoretical physics. His a ...
and
Evgenii M. Lifschitz Evgeny Mikhailovich Lifshitz (russian: Евге́ний Миха́йлович Ли́фшиц; February 21, 1915, Kharkiv, Russian Empire – October 29, 1985, Moscow, Russian SFSR) was a leading Soviet physicist and brother of the physicist ...
, ''The Classical Theory of Fields'', (1973) Pergammon Press, Oxford ''See section 87''. * Charles W. Misner,
Kip S. Thorne Kip Stephen Thorne (born June 1, 1940) is an American theoretical physics, theoretical physicist known for his contributions in gravitation, gravitational physics and astrophysics. A longtime friend and colleague of Stephen Hawking and Carl ...
, John Archibald Wheeler, '' Gravitation'', (1970) W.H. Freeman, New York; . * Bernard F. Schutz, ''A first course in general relativity'', (1985; 2002) Cambridge University Press: Cambridge, UK; . ''See chapter 6''. *
Robert M. Wald The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honou ...
, ''
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. ...
'', (1984) The University of Chicago Press, Chicago. ''See Section 3.3''.


References

{{Relativity General relativity Articles containing proofs