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In the special theory of relativity, **four-force** is a four-vector that replaces the classical force.

In special relativity, a **four-vector** is an object with four components, which transform in a specific way under Lorentz transformation. 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 (½,½) 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 physics, a **force** is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass to change its velocity, i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol **F**.

The four-force is defined as the rate of change in the four-momentum of a particle with respect to the particle's proper time:

In special relativity, **four-momentum** is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy *E* and three-momentum **p** = = *γm***v**, where **v** is the particle's three-velocity and γ the Lorentz factor, is

In relativity, **proper time** along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The **proper time interval** between two events on a world line is the change in proper time. This interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line. The proper time interval between two events depends not only on the events but also the world line connecting them, and hence on the motion of the clock between the events. It is expressed as an integral over the world line. An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (inertial) clock between the same two events. The twin paradox is an example of this effect.

- .

For a particle of constant invariant mass , where is the four-velocity, so we can relate the four-force with the four-acceleration as in Newton's second law:

The **invariant mass**, **rest mass**, **intrinsic mass**, **proper mass,** or in the case of bound systems simply **mass,** is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total energy and momentum that is the same in all frames of reference related by Lorentz transformations. If a center-of-momentum frame exists for the system, then the invariant mass of a system is equal to its total mass in that "rest frame". In other reference frames, where the system's momentum is nonzero, the total mass of the system is greater than the invariant mass, but the invariant mass remains unchanged.

In physics, in particular in special relativity and general relativity, a **four-velocity** is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space.

In the theory of relativity, **four-acceleration** is a four-vector that is analogous to classical acceleration. Four-acceleration has applications in areas such as the annihilation of antiprotons, resonance of strange particles and radiation of an accelerated charge.

- .

Here

and

where , and are 3-space vectors describing the velocity, the momentum of the particle and the force acting on it respectively.

From the formulae of the previous section it appears that the time component of the four-force is the power expended, , apart from relativistic corrections . This is only true in purely mechanical situations, when heat exchanges vanish or can be neglected.

If the full thermo-mechanical case, not only work, but also heat contributes to the change in energy, which is the time component of the energy–momentum covector. The time component of the four-force includes in this case a heating rate , besides the power .^{ [1] } Note that work and heat cannot be meaningfully separated, though, as they both carry inertia.^{ [2] } This fact extends also to contact forces, that is, to the stress-energy-momentum tensor ^{ [3] }^{ [2] }.

In thermodynamics, **work** performed by a system is energy transferred by the system to its surroundings, by a mechanism through which the system can spontaneously exert macroscopic forces on its surroundings, where those forces, and their external effects, can be measured. In the surroundings, through suitable passive linkages, the whole of the work done by such forces can lift a weight. Also, just through such mechanisms, energy can transfer from the surroundings to the system; in a sign convention used in physics, such energy transfer is counted as a negative amount of work done by the system on its surroundings.

Therefore, in thermo-mechanical situations the time component of the four-force is *not* proportional to the power but has a more generic expression, to be given case by case, which represents the supply of internal energy from the combination of work and heat,^{ [2] }^{ [1] }^{ [4] }^{ [3] } and which in the Newtonian limit becomes .

In general relativity the relation between four-force, and four-acceleration remains the same, but the elements of the four-force are related to the elements of the four-momentum through a covariant derivative with respect to proper time.

In addition, we can formulate force using the concept of coordinate transformations between different coordinate systems. Assume that we know the correct expression for force in a coordinate system at which the particle is momentarily at rest. Then we can perform a transformation to another system to get the corresponding expression of force.^{ [5] } In special relativity the transformation will be a Lorentz transformation between coordinate systems moving with a relative constant velocity whereas in general relativity it will be a general coordinate transformation.

Consider the four-force acting on a particle of mass which is momentarily at rest in a coordinate system. The relativistic force in another coordinate system moving with constant velocity , relative to the other one, is obtained using a Lorentz transformation:

where .

In general relativity, the expression for force becomes

with covariant derivative . The equation of motion becomes

where is the Christoffel symbol. If there is no external force, this becomes the equation for geodesics in the curved space-time. The second term in the above equation, plays the role of a gravitational force. If is the correct expression for force in a freely falling frame , we can use then the equivalence principle to write the four-force in an arbitrary coordinate :

In special relativity, Lorentz four-force (four-force acting to charged particle situated in electromagnetic field) can be expressed as:

- ,

where

- is the electromagnetic tensor,
- is the four-velocity, and
- is the electric charge.

In physics, the **Lorentz transformations** are a one-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parametrized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.

The **stress–energy tensor**, sometimes **stress–energy–momentum tensor** or **energy–momentum tensor**, is a tensor quantity in physics 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. The stress–energy tensor is the source 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 the mathematical field of differential geometry, the **Riemann curvature tensor** or **Riemann–Christoffel tensor** is the most common method used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold, that measures the extent to which the metric tensor is not locally isometric to that of Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In physics, a **wave vector** is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important. Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

In differential geometry, the **four-gradient** is the four-vector analogue of the gradient from vector calculus.

In a relativistic theory of physics, a **Lorentz scalar** is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of vectors, or from contracting tensors of the theory. While the components of vectors and tensors are in general altered under Lorentz transformations, Lorentz scalars remain unchanged.

In general relativity, a **geodesic** generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line 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.

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.

The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

The **Newman–Penrose** (**NP**) **formalism** is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

In mathematical physics, **spacetime algebra** (STA) is a name for the Clifford algebra Cℓ_{1,3}(**R**), or equivalently the geometric algebra G(M^{4}), which can be particularly closely associated with the geometry of special relativity and relativistic spacetime.

In theoretical physics, the composition of two non-collinear Lorentz boosts results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation. This rotation is called **Thomas rotation**, **Thomas–Wigner rotation** or **Wigner rotation**. The rotation was discovered by Llewellyn Thomas in 1926, and derived by Wigner in 1939. If a sequence of non-collinear boosts returns an object to its initial velocity, then the sequence of Wigner rotations can combine to produce a net rotation called the Thomas precession.

In theoretical physics, **relativistic Lagrangian mechanics** is Lagrangian mechanics applied in the context of special relativity and general relativity.

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 three-dimensional quantity in classical mechanics.

In physics, specifically general relativity, the **Mathisson–Papapetrou–Dixon equations** describe the motion of a massive spinning body moving in a gravitational field. Other equations with similar names and mathematical forms are the **Mathisson-Papapetrou equations** and **Papapetrou-Dixon equations**. All three sets of equations describe the same physics.

In mathematics and physics, **acceleration** is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of the "bend".

- 1 2 Grot, Richard A.; Eringen, A. Cemal (1966). "Relativistic continuum mechanics: Part I – Mechanics and thermodynamics".
*Int. J. Engng Sci*.**4**(6): 611–638, 664. doi:10.1016/0020-7225(66)90008-5. - 1 2 3 Eckart, Carl (1940). "The Thermodynamics of Irreversible Processes. III. Relativistic Theory of the Simple Fluid".
*Phys. Rev*.**58**(10): 919–924. Bibcode:1940PhRv...58..919E. doi:10.1103/PhysRev.58.919. - 1 2 C. A. Truesdell, R. A. Toupin:
*The Classical Field Theories*(in S. Flügge (ed.):*Encyclopedia of Physics, Vol. III-1*, Springer 1960). §§152–154 and 288–289. - ↑ Maugin, Gérard A. (1978). "On the covariant equations of the relativistic electrodynamics of continua. I. General equations".
*J. Math. Phys*.**19**(5): 1198–1205. Bibcode:1978JMP....19.1198M. doi:10.1063/1.523785. - ↑ Steven, Weinberg (1972).
*Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*. John Wiley & Sons, Inc. ISBN 0-471-92567-5.

- Rindler, Wolfgang (1991).
*Introduction to Special Relativity*(2nd ed.). Oxford: Oxford University Press. ISBN 0-19-853953-3.

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