Relativistic mechanics

In physics,[] relativistic mechanics refers to mechanics compatible with special relativity and general relativity. 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.
As with classical mechanics, the subject can be divided into "kinematics"; the description of motion by specifying positions, velocities and accelerations, and "dynamics"; a full description by considering energies, momenta, and angular momenta and their conservation laws, and forces acting on particles or exerted by particles. There is however a subtlety; what appears to be "moving" and what is "at rest"—which is termed by "statics" in classical mechanics—depends on the relative motion of observers who measure in frames of reference.
Some definitions and concepts from classical mechanics do carry over to SR, such as force as the time derivative of momentum, the work done by a particle as the line integral of force exerted on the particle along a path, and power as the time derivative of work done. However, there are a number of significant modifications to the remaining definitions and formulae. SR states that motion is relative and the laws of physics are the same for all experimenters irrespective of their inertial reference frames. In addition to modifying notions of space and time, SR forces one to reconsider the concepts of mass, momentum, and energy all of which are important constructs in Newtonian mechanics. SR shows that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated.
The equations become more complicated in the more familiar three-dimensional vector calculus formalism, due to the nonlinearity in the Lorentz factor, which accurately accounts for relativistic velocity dependence and the speed limit of all particles and fields. However, they have a simpler and elegant form in four-dimensional spacetime, which includes flat Minkowski space and curved spacetime, because three-dimensional vectors derived from space and scalars derived from time can be collected into four vectors, or four-dimensional tensors. The six-component angular momentum tensor is sometimes called a bivector because in the 3D viewpoint it is two vectors.

Relativistic kinematics

The relativistic four-velocity, that is the four-vector representing velocity in relativity, is defined as follows:
In the above, is the proper time of the path through spacetime, called the world-line, followed by the object velocity the above represents, and
is the four-position; the coordinates of an event. Due to time dilation, the proper time is the time between two events in a frame of reference where they take place at the same location. The proper time is related to coordinate time t by:
where is the Lorentz factor:
so it follows:
The first three terms, excepting the factor of, is the velocity as seen by the observer in their own reference frame. The is determined by the velocity between the observer's reference frame and the object's frame, which is the frame in which its proper time is measured. This quantity is invariant under Lorentz transformation, so to check to see what an observer in a different reference frame sees, one simply multiplies the velocity four-vector by the Lorentz transformation matrix between the two reference frames.

Relativistic dynamics

Rest mass and relativistic mass

The mass of an object as measured in its own frame of reference is called its rest mass or invariant mass and is sometimes written. If an object moves with velocity in some other reference frame, the quantity is often called the object's "relativistic mass" in that frame.
Some authors use to denote rest mass, but for the sake of clarity this article will follow the convention of using for relativistic mass and for rest mass.
Lev Okun has suggested that the concept of relativistic mass "has no rational justification today" and should no longer be taught.
Other physicists, including Wolfgang Rindler and T. R. Sandin, contend that the concept is useful.
See mass in special relativity for more information on this debate.
A particle whose rest mass is zero is called massless. Photons and gravitons are thought to be massless, and neutrinos are nearly so.

Relativistic energy and momentum

There are a couple of ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy, one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR.
The four-momentum of an object is straightforward, identical in form to the classical momentum, but replacing 3-vectors with 4-vectors:
The energy and momentum of an object with invariant mass, moving with velocity with respect to a given frame of reference, are respectively given by
The factor comes from the definition of the four-velocity described above. The appearance of may be stated in an alternative way, which will be explained in the next section.
The kinetic energy,, is defined as
and the speed as a function of kinetic energy is given by
The spatial momentum may be written as, preserving the form from Newtonian mechanics with relativistic mass substituted for Newtonian mass. However, this substitution fails for some quantities, including force and kinetic energy. Moreover, the relativistic mass is not invariant under Lorentz transformations, while the rest mass is. For this reason, many people prefer to use the rest mass and account for explicitly through the 4-velocity or coordinate time.
A simple relation between energy, momentum, and velocity may be obtained from the definitions of energy and momentum by multiplying the energy by, multiplying the momentum by, and noting that the two expressions are equal. This yields
may then be eliminated by dividing this equation by and squaring,
dividing the definition of energy by and squaring,
and substituting:
This is the relativistic energy–momentum relation.
While the energy and the momentum depend on the frame of reference in which they are measured, the quantity is invariant. Its value is times the squared magnitude of the 4-momentum vector.
The invariant mass of a system may be written as
Due to kinetic energy and binding energy, this quantity is different from the sum of the rest masses of the particles of which the system is composed. Rest mass is not a conserved quantity in special relativity, unlike the situation in Newtonian physics. However, even if an object is changing internally, so long as it does not exchange energy or momentum with its surroundings, its rest mass will not change and can be calculated with the same result in any reference frame.

Mass–energy equivalence

The relativistic energy–momentum equation holds for all particles, even for massless particles for which m₀ = 0. In this case:
When substituted into Ev = c²p, this gives v = c: massless particles always travel at the speed of light.
Notice that the rest mass of a composite system will generally be slightly different from the sum of the rest masses of its parts since, in its rest frame, their kinetic energy will increase its mass and their binding energy will decrease its mass. In particular, a hypothetical "box of light" would have rest mass even though made of particles which do not since their momenta would cancel.
Looking at the above formula for invariant mass of a system, one sees that, when a single massive object is at rest, there is a non-zero mass remaining: m₀ = E/''c''².
The corresponding energy, which is also the total energy when a single particle is at rest, is referred to as "rest energy". In systems of particles which are seen from a moving inertial frame, total energy increases and so does momentum. However, for single particles the rest mass remains constant, and for systems of particles the invariant mass remain constant, because in both cases, the energy and momentum increases subtract from each other, and cancel. Thus, the invariant mass of systems of particles is a calculated constant for all observers, as is the rest mass of single particles.

The mass of systems and conservation of invariant mass

For systems of particles, the energy–momentum equation requires summing the momentum vectors of the particles:
The inertial frame in which the momenta of all particles sums to zero is called the center of momentum frame. In this special frame, the relativistic energy–momentum equation has p = 0, and thus gives the invariant mass of the system as merely the total energy of all parts of the system, divided by c²
This is the invariant mass of any system which is measured in a frame where it has zero total momentum, such as a bottle of hot gas on a scale. In such a system, the mass which the scale weighs is the invariant mass, and it depends on the total energy of the system. It is thus more than the sum of the rest masses of the molecules, but also includes all the totaled energies in the system as well. Like energy and momentum, the invariant mass of isolated systems cannot be changed so long as the system remains totally closed, because the total relativistic energy of the system remains constant so long as nothing can enter or leave it.
An increase in the energy of such a system which is caused by translating the system to an inertial frame which is not the center of momentum frame, causes an increase in energy and momentum without an increase in invariant mass. E = m₀c², however, applies only to isolated systems in their center-of-momentum frame where momentum sums to zero.
Taking this formula at face value, we see that in relativity, mass is simply energy by another name. In 1927 Einstein remarked about special relativity, "Under this theory mass is not an unalterable magnitude, but a magnitude dependent on the amount of energy."