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 (Newton's second law), 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.
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.
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 (SR) and curved spacetime (GR), because three-dimensional vectors derived from space and scalars derived from time can be collected into four vectors, or four-dimensional tensors.
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.
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.
to denote rest mass, but for the sake of clarity this article will follow the convention of using
[2] Lev Okun has suggested that the concept of relativistic mass "has no rational justification today" and should no longer be taught.
There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws.
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.
, is defined as and the speed as a function of kinetic energy is given by The spatial momentum may be written as
Rest mass is not a conserved quantity in special relativity, unlike the situation in Newtonian physics.
In this case: When substituted into Ev = c2p, this gives v = c: massless particles (such as photons) always travel at the speed of light.
In particular, a hypothetical "box of light" would have rest mass even though made of particles which do not since their momenta would cancel.
In systems of particles which are seen from a moving inertial frame, total energy increases and so does momentum.
Taking this formula at face value, we see that in relativity, mass is simply energy by another name (and measured in different units).
In both cases, the mass difference between reactants and (cooled) products measures the mass of heat and light which will escape the reaction, and thus (using the equation) give the equivalent energy of heat and light which may be emitted if the reaction proceeds.
Historically, for example, Lise Meitner was able to use the mass differences in nuclei to estimate that there was enough energy available to make nuclear fission a favorable process.
The implications of this special form of Einstein's formula have thus made it one of the most famous equations in all of science.
Only when a transparent "window" was opened in the super-strong plasma-filled box, and light and heat were allowed to escape in a beam, and the bomb components to cool, would the system lose the mass associated with the energy of the blast.
If this heat and light were allowed to escape, the remains of the bomb would lose a gram of mass, as it cooled.
In special relativity, Newton's second law does not hold in the form F = ma, but it does if it is expressed as where p = γ(v)m0v is the momentum as defined above and m0 is the invariant mass.
Thus, the force is given by Starting from Carrying out the derivatives gives If the acceleration is separated into the part parallel to the velocity (a∥) and the part perpendicular to it (a⊥), so that: one gets By construction a∥ and v are parallel, so (v·a∥)v is a vector with magnitude v2a∥ in the direction of v (and hence a∥) which allows the replacement: then Consequently, in some old texts, γ(v)3m0 is referred to as the longitudinal mass, and γ(v)m0 is referred to as the transverse mass, which is numerically the same as the relativistic mass.
It should not be confused with the so-called four-force which is merely the 3-D force in the comoving frame of the object transformed as if it were a four-vector.
As with angular momentum, torque is additive, so for an extended object one sums or integrates over the distribution of mass.
The work-energy theorem says[12] the change in kinetic energy is equal to the work done on the body.