This article will use the Einstein summation convention.
The theory of general relativity required the adaptation of existing theories of physical, electromagnetic, and quantum effects to account for non-Euclidean geometries. These physical theories modified by general relativity are described below.
Classical mechanics and special relativity are lumped together here because special relativity is in many ways intermediate between general relativity and classical mechanics, and shares many attributes with classical mechanics.
In the following discussion, the mathematics of general relativity is used heavily. Also, under the principle of minimal coupling, the physical equations of special relativity can be turned into their general relativity counterparts by replacing the Minkowski metric (ηab) with the relevant metric of spacetime (gab) and by replacing any partial derivatives with covariant derivatives. In the discussions that follow, the change of metrics is implied.
Inertial motion is motion free of all forces. In Newtonian mechanics, the force F acting on a particle with mass m is given by Newton's second law, , where the acceleration is given by the second derivative of position r with respect to time t . Zero force means that inertial motion is just motion with zero acceleration:
The idea is the same in special relativity. Using Cartesian coordinates, inertial motion is described mathematically as:
where is the position coordinate and τ is proper time. (In Newtonian mechanics, τ ≡ t, the coordinate time).
In both Newtonian mechanics and special relativity, space and then spacetime are assumed to be flat, and we can construct a global Cartesian coordinate system. In general relativity, these restrictions on the shape of spacetime and on the coordinate system to be used are lost. Therefore, a different definition of inertial motion is required. In relativity, inertial motion occurs along timelike or null geodesics as parameterized by proper time. This is expressed mathematically by the geodesic equation:
where is a Christoffel symbol. Since general relativity describes four-dimensional spacetime, this represents four equations, with each one describing the second derivative of a coordinate with respect to proper time. In the case of flat space in Cartesian coordinates, we have , so this equation reduces to the special relativity form.
For gravitation, the relationship between Newton's theory of gravity and general relativity is governed by the correspondence principle: General relativity must produce the same results as gravity does for the cases where Newtonian physics has been shown to be accurate.
Around a spherically symmetric object, the Newtonian theory of gravity predicts that objects will be physically accelerated towards the center on the object by the rule
where G is Newton's Gravitational constant, M is the mass of the gravitating object, r is the distance to the gravitation object, and is a unit vector identifying the direction to the massive object.
In the weak-field approximation of general relativity, an identical coordinate acceleration must exist. For the Schwarzschild solution (which is the simplest possible spacetime surrounding a massive object), the same acceleration as that which (in Newtonian physics) is created by gravity is obtained when a constant of integration is set equal to 2MG/c2). For more information, see Deriving the Schwarzschild solution.
Some of the basic concepts of general relativity can be outlined outside the relativistic domain. In particular, the idea that mass/energy generates curvature in space and that curvature affects the motion of masses can be illustrated in a Newtonian setting.
General relativity generalizes the geodesic equation and the field equation to the relativistic realm in which trajectories in space are replaced with Fermi–Walker transport along world lines in spacetime. The equations are also generalized to more complicated curvatures.
The basic structure of general relativity, including the geodesic equation and Einstein field equation, can be obtained from special relativity by examining the kinetics and dynamics of a particle in a circular orbit about the earth. In terms of symmetry, the transition involves replacing global Lorentz covariance with local Lorentz covariance.
In classical mechanics, conservation laws for energy and momentum are handled separately in the two principles of conservation of energy and conservation of momentum. With the advent of special relativity, these two conservation principles were united through the concept of mass-energy equivalence.
Mathematically, the general relativity statement of energy–momentum conservation is:
where is the stress–energy tensor, the comma indicates a partial derivative and the semicolon indicates a covariant derivative. The terms involving the Christoffel symbols are absent in the special relativity statement of energy–momentum conservation.
Unlike classical mechanics and special relativity, it is not usually possible to unambiguously define the total energy and momentum in general relativity, so the tensorial conservation laws are local statements only (see ADM energy, though). This often causes confusion in time-dependent spacetimes which apparently do not conserve energy, although the local law is always satisfied. Exact formulation of energy–momentum conservation on an arbitrary geometry requires use of a non-unique stress–energy–momentum pseudotensor.
General relativity modifies the description of electromagnetic phenomena by employing a new version of Maxwell's equations. These differ from the special relativity form in that the Christoffel symbols make their presence in the equations via the covariant derivative.
The source equations of electrodynamics in curved spacetime are (in cgs units)
where Fab is the electromagnetic field tensor representing the electromagnetic field and Ja is a four-current representing the sources of the electromagnetic field.
The source-free equations are the same as their special relativity counterparts.
The effect of an electromagnetic field on a charged object is then modified to
where q is the charge on the object, m is the rest mass of the object and P a is the four-momentum of the charged object. Maxwell's equations in flat spacetime are recovered in rectangular coordinates by reverting the covariant derivatives to partial derivatives. For Maxwell's equations in flat spacetime in curvilinear coordinates see or
A force is an influence that can cause an object to change its velocity unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity. The SI unit of force is the newton (N), and force is often represented by the symbol F.
In physics, specifically in electromagnetism, the Lorentz force law is the combination of electric and magnetic force on a point charge due to electromagnetic fields. The Lorentz force, on the other hand, is a physical effect that occurs in the vicinity of electrically neutral, current-carrying conductors causing moving electrical charges to experience a magnetic force.
In Newtonian mechanics, momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If m is an object's mass and v is its velocity, then the object's momentum p is: In the International System of Units (SI), the unit of measurement of momentum is the kilogram metre per second (kg⋅m/s), which is dimensionally equivalent to the newton-second.
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper, On the Electrodynamics of Moving Bodies, the theory is presented as being based on just two postulates:
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 tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.
In special relativity, a four-vector is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.
In the special theory of relativity, four-force is a four-vector that replaces the classical force.
Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference. Galileo Galilei first described this principle in 1632 in his Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer below the deck would not be able to tell whether the ship was moving or stationary.
In the general theory of relativity, the Einstein field equations relate the geometry of spacetime to the distribution of matter within it.
When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity.
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of moving objects are comparable to the speed of light c. As a result, classical mechanics is extended correctly to particles traveling at high velocities and energies, and provides a consistent inclusion of electromagnetism with the mechanics of particles. This was not possible in Galilean relativity, where it would be permitted for particles and light to travel at any speed, including faster than light. The foundations of relativistic mechanics are the postulates of special relativity and general relativity. The unification of SR with quantum mechanics is relativistic quantum mechanics, while attempts for that of GR is quantum gravity, an unsolved problem in physics.
In general relativity, if two objects are set in motion along two initially parallel trajectories, the presence of a tidal gravitational force will cause the trajectories to bend towards or away from each other, producing a relative acceleration between the objects.
Fermi–Walker transport is a process in general relativity used to define a coordinate system or reference frame such that all curvature in the frame is due to the presence of mass/energy density and not due to arbitrary spin or rotation of the frame. It was discovered by Fermi in 1921 and rediscovered by Walker in 1932.
A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the Earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism.
In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation.
The mathematics of general relativity is complicated. 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 solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches the speed of light, meaning that more variables and more complicated mathematics are required to calculate the object's motion. As a result, relativity requires the use of concepts such as vectors, tensors, pseudotensors and curvilinear coordinates.
There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.
Newton–Cartan theory is a geometrical re-formulation, as well as a generalization, of Newtonian gravity first introduced by Élie Cartan and Kurt Friedrichs and later developed by G. Dautcourt, W. G. Dixon, P. Havas, H. Künzle, Andrzej Trautman, and others. In this re-formulation, the structural similarities between Newton's theory and Albert Einstein's general theory of relativity are readily seen, and it has been used by Cartan and Friedrichs to give a rigorous formulation of the way in which Newtonian gravity can be seen as a specific limit of general relativity, and by Jürgen Ehlers to extend this correspondence to specific solutions of general relativity.
Classical mechanics is a physical theory describing the motion of objects such as projectiles, parts of machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics involved substantial change in the methods and philosophy of physics. The qualifier classical distinguishes this type of mechanics from physics developed after the revolutions in physics of the early 20th century, all of which revealed limitations in classical mechanics.
In theoretical physics, relativistic Lagrangian mechanics is Lagrangian mechanics applied in the context of special relativity and general relativity.