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The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total energy and momentum that is the same in all frames of reference related by Lorentz transformations. [1] If a center-of-momentum frame exists for the system, then the invariant mass of a system is equal to its total mass in that "rest frame". In other reference frames, where the system's momentum is nonzero, the total mass (a.k.a. relativistic mass) of the system is greater than the invariant mass, but the invariant mass remains unchanged.
Because of mass–energy equivalence, the rest energy of the system is simply the invariant mass times the speed of light squared. Similarly, the total energy of the system is its total (relativistic) mass times the speed of light squared.
Systems whose four-momentum is a null vector (for example, a single photon or many photons moving in exactly the same direction) have zero invariant mass and are referred to as massless . A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized tachyon), and these do not appear to exist. Any time-like four-momentum possesses a reference frame where the momentum (3-dimensional) is zero, which is a center of momentum frame. In this case, invariant mass is positive and is referred to as the rest mass.
If objects within a system are in relative motion, then the invariant mass of the whole system will differ from the sum of the objects' rest masses. This is also equal to the total energy of the system divided by c 2. See mass–energy equivalence for a discussion of definitions of mass. Since the mass of systems must be measured with a weight or mass scale in a center of momentum frame in which the entire system has zero momentum, such a scale always measures the system's invariant mass. For example, a scale would measure the kinetic energy of the molecules in a bottle of gas to be part of invariant mass of the bottle, and thus also its rest mass. The same is true for massless particles in such system, which add invariant mass and also rest mass to systems, according to their energy.
For an isolated massive system, the center of mass of the system moves in a straight line with a steady subluminal velocity (with a velocity depending on the reference frame used to view it). Thus, an observer can always be placed to move along with it. In this frame, which is the center-of-momentum frame, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which exists under these assumptions, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by c2. This total energy in the center of momentum frame, is the minimum energy which the system may be observed to have, when seen by various observers from various inertial frames.
Note that for reasons above, such a rest frame does not exist for single photons, or rays of light moving in one direction. When two or more photons move in different directions, however, a center of mass frame (or "rest frame" if the system is bound) exists. Thus, the mass of a system of several photons moving in different directions is positive, which means that an invariant mass exists for this system even though it does not exist for each photon.
The invariant mass of a system includes the mass of any kinetic energy of the system constituents that remains in the center of momentum frame, so the invariant mass of a system may be greater than sum of the invariant masses (rest masses) of its separate constituents. For example, rest mass and invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass).
Consider the simple case of two-body system, where object A is moving towards another object B which is initially at rest (in any particular frame of reference). The magnitude of invariant mass of this two-body system (see definition below) is different from the sum of rest mass (i.e. their respective mass when stationary). Even if we consider the same system from center-of-momentum frame, where net momentum is zero, the magnitude of the system's invariant mass is not equal to the sum of the rest masses of the particles within it.
The kinetic energy of such particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and both terms contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in the center of momentum frame (again, called the "rest frame" if the system is bound).
They will often also interact through one or more of the fundamental forces, giving them a potential energy of interaction, possibly negative.
In particle physics, the invariant massm0 is equal to the mass in the rest frame of the particle, and can be calculated by the particle's energy E and its momentum p as measured in any frame, by the energy–momentum relation:
or in natural units where c = 1,
This invariant mass is the same in all frames of reference (see also special relativity). This equation says that the invariant mass is the pseudo-Euclidean length of the four-vector (E, p), calculated using the relativistic version of the Pythagorean theorem which has a different sign for the space and time dimensions. This length is preserved under any Lorentz boost or rotation in four dimensions, just like the ordinary length of a vector is preserved under rotations. In quantum theory the invariant mass is a parameter in the relativistic Dirac equation for an elementary particle. The Dirac quantum operator corresponds to the particle four-momentum vector.
Since the invariant mass is determined from quantities which are conserved during a decay, the invariant mass calculated using the energy and momentum of the decay products of a single particle is equal to the mass of the particle that decayed. The mass of a system of particles can be calculated from the general formula:
where
The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") W of the reaction is defined as follows (in natural units):
If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle.
In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the missing energy) the transverse mass is used.
In a two-particle collision (or a two-particle decay) the square of the invariant mass (in natural units) is
The invariant mass of a system made of two massless particles whose momenta form an angle has a convenient expression:
In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle and pseudorapidity . Additionally the transverse momentum, , is usually measured. In this case if the particles are massless, or highly relativistic () then the invariant mass becomes:
Rest energy (also called rest mass energy) is the energy associated with a particle's invariant mass. [2] [3]
The rest energy of a particle is defined as:
where is the speed of light in vacuum. [2] [3] [4] In general, only differences in energy have physical significance. [5]
The concept of rest energy follows from the special theory of relativity that leads to Einstein's famous conclusion about equivalence of energy and mass. See Special relativity § Relativistic dynamics and invariance .
In physics, angular momentum is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.
Compton scattering is the quantum theory of high frequency photons scattering following an interaction with a charged particle, usually an electron. Specifically, when the photon hits electrons, it releases loosely bound electrons from the outer valence shells of atoms or molecules.
In physics, an elastic collision is an encounter (collision) between two bodies in which the total kinetic energy of the two bodies remains the same. In an ideal, perfectly elastic collision, there is no net conversion of kinetic energy into other forms such as heat, noise, or potential energy.
Pair production is the creation of a subatomic particle and its antiparticle from a neutral boson. Examples include creating an electron and a positron, a muon and an antimuon, or a proton and an antiproton. Pair production often refers specifically to a photon creating an electron–positron pair near a nucleus. As energy must be conserved, for pair production to occur, the incoming energy of the photon must be above a threshold of at least the total rest mass energy of the two particles created. Conservation of energy and momentum are the principal constraints on the process. All other conserved quantum numbers of the produced particles must sum to zero – thus the created particles shall have opposite values of each other. For instance, if one particle has electric charge of +1 the other must have electric charge of −1, or if one particle has strangeness of +1 then another one must have strangeness of −1.
In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass cannot change, so the quantity can neither be added nor be removed. Therefore, the quantity of mass is conserved over time.
In special relativity, four-momentum (also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy E and three-momentum p = (px, py, pz) = γmv, where v is the particle's three-velocity and γ the Lorentz factor, is
In special relativity, a four-vector is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.
The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find.
The word "mass" has two meanings in special relativity: invariant mass is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity of the observer. According to the concept of mass–energy equivalence, invariant mass is equivalent to rest energy, while relativistic mass is equivalent to relativistic energy.
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). 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.
In experimental particle physics, pseudorapidity, , is a commonly used spatial coordinate describing the angle of a particle relative to the beam axis. It is defined as
In theoretical physics, the Weinberg–Witten (WW) theorem, proved by Steven Weinberg and Edward Witten, states that massless particles (either composite or elementary) with spin j > 1/2 cannot carry a Lorentz-covariant current, while massless particles with spin j > 1 cannot carry a Lorentz-covariant stress-energy. The theorem is usually interpreted to mean that the graviton (j = 2) cannot be a composite particle in a relativistic quantum field theory.
In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy to invariant mass and momentum. It is the extension of mass–energy equivalence for bodies or systems with non-zero momentum. It can be written as the following equation:
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.
In physics, the center-of-momentum frame, also known as zero-momentum frame, is the inertial frame in which the total momentum of the system vanishes. It is unique up to velocity, but not origin.
In particle physics, particle decay is the spontaneous process of one unstable subatomic particle transforming into multiple other particles. The particles created in this process must each be less massive than the original, although the total mass of the system must be conserved. A particle is unstable if there is at least one allowed final state that it can decay into. Unstable particles will often have multiple ways of decaying, each with its own associated probability. Decays are mediated by one or several fundamental forces. The particles in the final state may themselves be unstable and subject to further decay.
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, Mécanique analytique.
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non commuting symmetry operators or that the non degenerate states are also eigenvectors of symmetry operators.