# Four-force

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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.

## In special relativity

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 = = γmv, 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.

$\mathbf {F} ={\mathrm {d} \mathbf {P} \over \mathrm {d} \tau }$ .

For a particle of constant invariant mass $m>0$ , $\mathbf {P} =m\mathbf {U}$ where $\mathbf {U} =\gamma (c,\mathbf {u} )$ is the four-velocity, so we can relate the four-force with the four-acceleration $\mathbf {A}$ 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.

$\mathbf {F} =m\mathbf {A} =\left(\gamma {\mathbf {f} \cdot \mathbf {u} \over c},\gamma {\mathbf {f} }\right)$ .

Here

${\mathbf {f} }={\mathrm {d} \over \mathrm {d} t}\left(\gamma m{\mathbf {u} }\right)={\mathrm {d} \mathbf {p} \over \mathrm {d} t}$ and

${\mathbf {f} \cdot \mathbf {u} }={\mathrm {d} \over \mathrm {d} t}\left(\gamma mc^{2}\right)={\mathrm {d} E \over \mathrm {d} t}.$ where $\mathbf {u}$ , $\mathbf {p}$ and $\mathbf {f}$ are 3-space vectors describing the velocity, the momentum of the particle and the force acting on it respectively.

## Including thermodynamic interactions

From the formulae of the previous section it appears that the time component of the four-force is the power expended, $\mathbf {f} \cdot \mathbf {u}$ , apart from relativistic corrections $\gamma /c$ . 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 $h$ , besides the power $\mathbf {f} \cdot \mathbf {u}$ .  Note that work and heat cannot be meaningfully separated, though, as they both carry inertia.  This fact extends also to contact forces, that is, to the stress-energy-momentum tensor   . 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 $\mathbf {f} \cdot \mathbf {u}$ 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,     and which in the Newtonian limit becomes $h+\mathbf {f} \cdot \mathbf {u}$ .

## In general relativity

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.

$F^{\lambda }:={\frac {DP^{\lambda }}{d\tau }}={\frac {dP^{\lambda }}{d\tau }}+\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }P^{\nu }$ 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.  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 $F^{\mu }=(F^{0},\mathbf {F} )$ acting on a particle of mass $m$ which is momentarily at rest in a coordinate system. The relativistic force $f^{\mu }$ in another coordinate system moving with constant velocity $v$ , relative to the other one, is obtained using a Lorentz transformation:

${\mathbf {f} }={\mathbf {F} }+(\gamma -1){\mathbf {v} }{{\mathbf {v} }\cdot {\mathbf {F} } \over v^{2}},$ $f^{0}=\gamma {\boldsymbol {\beta }}\cdot \mathbf {F} ={\boldsymbol {\beta }}\cdot \mathbf {f} .$ where ${\boldsymbol {\beta }}=\mathbf {v} /c$ .

In general relativity, the expression for force becomes

$f^{\mu }=m{DU^{\mu } \over d\tau }$ with covariant derivative $D/d\tau$ . The equation of motion becomes

$m{d^{2}x^{\mu } \over d\tau ^{2}}=f^{\mu }-m\Gamma _{\nu \lambda }^{\mu }{dx^{\nu } \over d\tau }{dx^{\lambda } \over d\tau },$ where $\Gamma _{\nu \lambda }^{\mu }$ 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 $f_{f}^{\alpha }$ is the correct expression for force in a freely falling frame $\xi ^{\alpha }$ , we can use then the equivalence principle to write the four-force in an arbitrary coordinate $x^{\mu }$ :

$f^{\mu }={\partial x^{\mu } \over \partial \xi ^{\alpha }}f_{f}^{\alpha }.$ ## Examples

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

$F_{\mu }=qF_{\mu \nu }U^{\nu }$ ,

where

• $F_{\mu \nu }$ is the electromagnetic tensor,
• $U^{\nu }$ is the four-velocity, and
• $q$ is the electric charge.

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