Relativistic mechanics

Relativistic mechanics is a branch of physics used to describe systems of particles or fluids. It is specifically required when objects move at velocities comparable to the speed of light, denoted as *c*. While classical mechanics works for everyday speeds, it fails when things move extremely fast. Relativistic mechanics extends these rules to include high energies and electromagnetism. It is built upon the two main foundations of special relativity (SR) and general relativity (GR). Special relativity deals with motion in flat spacetime, while general relativity describes motion in curved spacetime.

The subject is organized into two primary fields: kinematics and dynamics. Kinematics is the description of motion by specifying positions, velocities, and accelerations. Dynamics provides a fuller description by considering forces, energies, momenta, and angular momenta. A key concept here is the frame of reference. What appears to be "moving" or "at rest" depends entirely on the relative motion of the observers. In classical mechanics, this is called statics, but in relativity, the observer's perspective is central to the measurement.

To understand how this works, we must look at the four-dimensional view of the universe. In three-dimensional vector calculus, equations can become very complicated. This is due to the nonlinearity of the Lorentz factor, which accounts for the speed limit of all particles and fields. However, the math becomes much more elegant in four-dimensional spacetime. In this view, three-dimensional space and one dimension of time are collected into four-vectors or four-dimensional tensors. One such example is the relativistic four-velocity. This represents velocity as a path through spacetime, known as a world-line.

Relativistic mechanics also changes how we define mass. An object has a "rest mass" or "invariant mass" when measured in its own frame of reference. This is written as *m₀*. When an object moves in another frame, some call this quantity "relativistic mass." This has caused a debate among physicists. In 2008, the concept was viewed differently by different experts. Lev Okun suggested that relativistic mass has no rational justification and should not be taught. However, other physicists like Wolfgang Rindler and T. R. Sandin argue that the concept remains useful.

Mass and energy are deeply connected through the energy-momentum relation. In special relativity, the Newtonian definitions of momentum and energy do not work because they are not conserved. To fix this, scientists use new definitions that account for relativistic speeds. This results in the four-momentum, which uses four-vectors instead of standard three-vectors. The energy of an object depends on its invariant mass and its velocity. A very important result is that massless particles, such as photons or gravitons, must always travel at the speed of light. This is because their rest mass is zero.

One of the most famous parts of this science is mass-energy equivalence. This is expressed by the relation *E = mc²*. This formula shows that mass is not an unalterable magnitude. As Albert Einstein remarked in 1927, mass is actually identical to the amount of energy. For a single particle at rest, the energy is called its rest energy. For a system of many particles, the invariant mass is not just the sum of their individual rest masses. It also includes their kinetic energy and their binding energy.

We can see these rules in action by looking at closed systems. An isolated or "totally-closed" system is one where no mass or energy can enter or leave. In such a system, the total energy, momentum, and invariant mass are all conserved. If a system is not closed and loses energy as heat or light, its mass will actually decrease. This is because the energy leaving the system is tied to the mass it holds. This connection explains why the mass of a bottle of hot gas on a scale is slightly more than the sum of its molecules. The scale is weighing the total energy of the system.