Four-vector

A four-vector, a concept of relativity, is a vector in a real, four-dimensional space with an indefinite length squared. In two relatively moving inertial systems the components of the four-vector related by a Lorentz transformation along with each other. For example, the time and spatial coordinates of an event in space-time the components of a four-vector, as are the energy and momentum of a particle, the components of a four vector.

Spelling

We used the abbreviations for the contravariant and covariant representation for a four-vector. It Greek indices are usually used when the values ​​go through 0,1,2,3. The letters in the theory of relativity are written preferred.

Position vector

The local vector or local four-vector of a particle includes both the time coordinate and the spatial coordinates of an event. The time coordinate is multiplied in the theory of relativity the speed of light, so that it has as the spatial coordinates of the dimension of a length.

The contravariant representation of the local four-vector

That is a four-vector, it follows that he is a coordinate vector to an orthonormal basis of the Minkowski space and varies accordingly contravariant means of a Lorentz transformation with base change.

In the metric of relativity is the time coordinate of the opposite sign of the three spatial coordinates, so the metric is the signature ( --- ) or (- ). Especially in texts on special relativity the first signature is mainly used, but this is just a convention, and varies depending on the author. For special processes, such as the entry into a black hole, change the sign in the metric that describes the black hole (eg, the Schwarzschild metric ) - space and time swap their importance.

Derived four-vectors

From the local four-vector for additional four-vectors can be derived and define.

Four-velocity

The four-vector of the velocity is given by differentiation of the position vector after the proper time.

The proper time is defined by the time dilation as

Where the Lorentz factor. This results in the four-velocity to

Interpretation

The norm of the four-velocity is apparent both in the special as well as in the general theory of relativity to

In other words, every object always moves with the speed of light through the four dimensions of space-time. This result explains the time dilation as follows: If an object from a reference system considered at rest, so it moves with the speed of light in the direction of the time dimension. If this subject, however, accelerated in space, its motion must be parallel to the slow time (time flow is slowed down ) so that the norm of the four-velocity remains constant. But since slowed down the flow of time, the rate appears to be increased in the four-vector.

Photons and other massless particles always move at the speed of light through space and rest it in time ( four-velocity is not defined ). Would move faster than light through space an object, it would have to have an imaginary velocity in the time to " compensate " the surplus.

Four-momentum

The four-momentum is defined analogously to the classical momentum as

The mass of the body. With the equivalence of mass and energy of the four-momentum can be used as

Be written. Here is the relativistic momentum, which differs from the classical momentum vector by the factor. Since the four-momentum combines the energy and the spatial impulse, he is also called the energy-momentum vector.

From the quadrature of the four-momenta of the energy - momentum relation yields

From the time-and location-independent Hamiltonian function for free relativistic particles can be derived.

Four acceleration

Again to derive the four-velocity is obtained by the four- acceleration.

The 0 -th component of the four- acceleration is determined to

Are the spatial components of the four- acceleration

Overall, we obtain for the four acceleration the result

The four acceleration consists of a part with a factor and a part. Therefore obtained four different accelerations for parallel and perpendicular acceleration. With the Grassmann identity

One can transform an expression for the spatial part of the four-vector. It is observed that

Is. It follows

And thus a total of

Four-force and equation of motion

As with the four-momentum can be a four-force, also called Minkowski force, analogue to define the corresponding Newtonian force.

This is the equation of motion of special relativity. It describes accelerated motions in an inertial system.

Further, the four-force can be related to the Newtonian force in relation:

In the inertial frame in which the mass approximately at rest ( calm at the time, then for sufficiently small because the acceleration is limited ), the classical Newtonian equation must apply:

The Newtonian force and the spatial part of the four-force is.

Valid in any inertial frame

That is, the space portion of the Minkowski force is the Newtonian force, the parallel to the velocity component is multiplied. is transmitted by the acceleration performance.

Co-and contravariant vectors

The components of a contravariant four-vector under Lorentz transformations go in

About. It writes its components with the above numbers.

Below indices denote components of a covariant four-vector with the contragredient ( opposite ) transformation law

The two transformation laws are not equal, but equivalent, because Lorentz transformations satisfy by definition

Therefore, yields the components of the covariant vector

Associated with the contravariant vector. is the usual Minkowski metric of the SRT.

For example, the partial derivatives of a function are the components of a covariant vector. Lorentz transformations form up down and define the transformed function by the requirement that the transformed function on the transformed site have the same value as the original function to the original location

The partial derivatives transform kontragredient because of the chain rule,