Four-momentum

As a four-momentum of a particle is defined in relativistic physics summarize its conservation laws of energy and momentum in the form of a four-vector.

Energy - momentum relation

In units where the speed of light has the dimensionless value, hanging by the energy and momentum of a particle of mass with its velocity

Together. The length of the square of the four-momentum is - like any scalar / dot product of any four-vectors - invariant under Lorentz transformation. That is, it is independent of the speed is always equal to the square of the mass:

This relationship is called the energy / momentum relation.

Derivation of the velocity dependence of the energy and momentum

As the energy and momentum of a particle of mass on its velocity depend on results in the theory of relativity from the fact that energy and momentum for each observer are additive conserved quantities.

We refer to them collectively with. If a particle plays an additive conserved quantity and other particles, the conserved quantity, then the system of the two particles leads to the conserved quantity.

Also a moving observer is both particles and conserved quantities fixed, but they have not necessarily the same, but transformed values ​​. However, it must be true that the sum of these values ​​is the transform of the sum,

Also comes (for all figures ) a replicated system with a conserved quantity for the moving observer, the multiplied conserved quantity

About. The states mathematically that the conserved quantities, which measures a moving observer, by a linear transformation

Related to the conservation values ​​of the stationary observer.

The linear transformation is limited by the fact that such an equation must hold for each pair of observers, the reference systems of the observers emerge by Lorentz transformations and shifts apart. Slopes of the reference systems of the first and second observer by and from the second to a third through together, then the reference system depends from the first to the third through together. Just need the corresponding transformations of the conserved quantities

. meet

In the simplest case. Since Lorentz transformations matrices are, therefore relates to the simplest non-trivial transformation law, and not just a true, four conserved quantities, as the four-vector, transform, such as the space-time coordinates,

In anticipation of the results of our analysis, we call this the four-momentum four-vector.

In particular, a particle at rest does not change during turns. Therefore, do not change those components of its four-momentum, which merge as a three-dimensional position vector under rotations in a rotated vector. However, the only such vector is the zero vector. So the four-momentum of a particle at rest has a value

The name is chosen in anticipation of the later result, is here but first for any value.

For a along the axis moving observer, the particles have a Lorentz transform of four-momentum has a speed and ( for simplicity, we expect measurement systems with )

Expanding the four conserved quantities on the speed

And compared with Newtonian mechanics, then reveals the physical meaning of the components of the four-momentum: The first component is the energy of the three components that change under rotations as a position vector of the pulse are

As in Newtonian mechanics is known as the speed-independent parameter in the ratio indicating the momentum of a particle as a function of its speed, the mass. You must be positive all observations after.

Viewed in SI units

The energy is when we insert the conventional factors,

It is bounded from below and subject to minimum rest

The pulse is

The relative energy - momentum relation can be derived with the help of skilful addition of a "zero" directly from the square of the energy. The assignment is:

Splits to the mass of the four-momentum from, the four-velocity remains that is the derivative of the world line that passes through the particles, according to his own time:

At this point it is emphasized with respect to the first equation, that m in contrast to the four vectors p, and u is a scalar factor. Also the differential of proper time is - in contrast to - a scalar quantity and the denominator.

The four-velocity is the normalized tangent vector to the world line,

Other conserved quantities, the angular momentum and the initial energy focus, transform under a six-dimensional representation of the Lorentz transformations.

Application: equation of motion and the force / power four-vector

In the comoving system is and remains zero as long as no force is applied. However, if a force is applied for a time and at the same time an external power L is supplied to increase both the speed and the energy of the particle (in the same reference system as before! ), And indeed is considered by the impulse and the power supply as the equation of motion: it will therefore, inter alia, the rest energy of the system of mc2 mc2 L on δτ increases (ie, the mass is slightly increased; see equivalence of mass and energy). The right-hand side of this equation defines the power-efficiency of four vector. Same time, through the impulse speed - and thus the kinetic energy - increased. It is assumed that the zero-based rate after the increase is still small compared to the speed of light remains, so that in the comoving system, Newtonian physics is valid.