The total energy is determined by approximate solutions of the time-dependent Schrödinger equation, usually with no relativistic terms included, and by making use of the Born – Oppenheimer approximation, which allows for the separation of electronic and nuclear motions, thereby simplifying the Schrödinger equation.

The important fact is that at very high velocities the behavior of the particle-like modes becomes distinct from the relativistic one-they can reach the speed of light limit at finite energy ; also the faster-than-light propagation is possible without requiring moving objects to have imaginary mass.

In a sphere of radius R, the entropy in a relativistic gas increases as the energy increases.

This same equation also connects relativistic mass and " relativistic energy " ( total system energy ).

The latter two " relativistic " mass and energy are concepts that are related to their " rest " counterparts, but they do not have the same value as their rest counterparts in systems where there is a net momentum.

then the relativistic fractional binding energy of a neutron star is

Using relativistic relations for energy and momentum, we have

The second shortcoming, related to the first, is that in quantum mechanics there is no mechanism to describe particle creation and annihilation ; this is crucial for describing phenomena such as pair production which result from the conversion between mass and energy according to the relativistic relation E = mc < sup > 2 </ sup >.

The chemical properties of rutherfordium were based on calculation which indicated that the relativistic effects on the electron shell might be strong enough that the p orbitals have a lower energy level than the d orbitals, therefore making the element behave more like lead.

The extension of Rutherford-type scattering to energy regions in which the incoming particle has spin and magnetic moment, and is traveling at relativistic energies, and there is enough momentum-transfer that the struck particle recoils with some of the incoming particle's energy ( so the process is inelastic rather than elastic ), is called Mott scattering.

As relativistic electrons travel through a medium, they lose energy producing a cone of blue light through the Cerenkov effect, and it is this light that is directly detected.

Other experiments include, for instance, relativistic energy and momentum increase at high velocities, time dilation of moving particles, and modern searches for Lorentz violations.

( where p is the relativistic momentum of the bradyon and m is its rest mass ) should still apply, along with the formula for the total energy of a particle:

Theoretically attainable momentum with which the ship can launch a helium atom can be obtained from the well-known relativistic relation between the mass, energy and momentum:

One could, for example, formally take the relativistic expression for the energy

Thus for example atomic processes that are observed as rates, will necessarily be adjusted in a way consistent with relativity, while those involving the measurement of energy and momentum, which themselves form a relativistic vector, will undergo parallel adjustment which preserves the relativistic covariance of the observed values.

In this model, pre-existing low-energy photons are scattered by relativistic electrons within the explosion, augmenting their energy by a large factor and transforming them into gamma-rays.

After scattering, the possibility that the electron might be accelerated to a significant fraction of the speed of light, requires that its total energy be represented using the relativistic energy-momentum relation:

The lower energy photons produced from this spectrum are scattered to higher energies by relativistic electrons in the surrounding corona.

Temperatures were so high that the random motions of particles were at relativistic speeds, and particle – antiparticle pairs of all kinds were being continuously created and destroyed in collisions.

* Tensor – vector – scalar gravity ( TeVeS ) ( 2004 ), a relativistic modification of MOND by Jacob Bekenstein

( The other four in the modern interpretation, different from the Greeks ' ideas, are: baryonic matter ; radiation – photons and the highly relativistic neutrinos, which may be considered hot dark matter ; cold dark matter ; and the term due to spatial curvature – loosely, gravitational self-energy.

Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein – Gordon equation, and describes a spinless particle field ( e. g. pi meson ).

The Navier – Stokes equations assume that the fluid being studied is a continuum ( it is infinitely divisible and not composed of particles such as atoms or molecules ), and is not moving at relativistic velocities.

There has been work in developing relativistic versions of de Broglie – Bohm theory.

In 1996, Partha Ghose had presented a relativistic quantum mechanical description of spin-0 and spin-1 bosons starting from the Duffin – Kemmer – Petiau equation, setting out Bohmian trajectories for massive bosons and for massless bosons, thus also for photons.

* 1982 – Joseph Taylor and Joel Weisberg show that the rate of energy loss from the binary pulsar PSR B1913 + 16 agrees with that predicted by the general relativistic quadrupole formula to within 5 %

The time – time component is the density of relativistic mass, i. e. the energy density divided by the speed of light squared.

Chemical studies performed in 2007 – 2008 indicate that flerovium is unexpectedly volatile for a group 14 element ; in preliminary results it even seemed to exhibit noble-gas-like properties due to relativistic effects.

A universe where the very laws of nature are different – for example, one in which there are no relativistic limitations and the speed of light can be exceeded – would in general count as a parallel universe but not an alternative reality.

Another, similar, alternative is the relativistic modified gravity ( MOG ) theory, also called scalar – tensor – vector gravity ( STVG ), of John Moffat.

Everett claims that the universe has a single quantum state, which he called the universal wavefunction, that always evolves according to the Schrödinger equation or some relativistic equivalent ; now the measurement problem suggests the universal wavefunction will be in a superposition corresponding to many different definite macroscopic realms (" macrorealms "); that one can recover the subjective appearance of a definite macrorealm by postulating that all the various definite macrorealms are actual – it seems to each observer that " we just happen to be in one rather than the others " because " we " are in all of them, but each are mutually unobservable.

Tensor – vector – scalar gravity ( TeVeS ) is a proposed relativistic theory that is equivalent to Modified Newtonian dynamics ( MOND ) in the non-relativistic limit, which purports to explain the galaxy rotation problem without invoking dark matter.

The Klein – Gordon equation ( Klein – Fock – Gordon equation or sometimes Klein – Gordon – Fock equation ) is a relativistic version of the Schrödinger equation.

In the language of four-vectors and tensors the angular momentum of a particle in relativistic mechanics is expressed as an antisymmetric tensor of second order

A relativistic treatment results in a momentum increase of about 1 part in 37, 000 for the electron.

Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, is approximately equal to, the Newtonian expression for momentum.

We also note that a two-body Dirac equation composed of a Dirac operator for each of the two point particles interacting via the Coulomb interaction can be exactly separated in the ( relativistic ) center of momentum frame and the resulting ground state eigenvalue has been obtained very accurately using the Finite element methods of J. Shertzer.

Due to relativistic effects ( Dirac equation ), there is a magnetic momentum, μ < sub > S </ sub >, arising from the electron spin

The flux of relativistic mass across the x < sup > i </ sup > surface is equivalent to the density of the i < sup > th </ sup > component of linear momentum,

The invariant mass is the relativistic mass of the system when viewed in the center of momentum frame.

In the same year, Majorana published his paper on a relativistic theory of particles with arbitrary intrinsic momentum, in which he developed and applied infinite dimensional representations of the Lorentz group, and gave a theoretical basis for the mass spectrum of elementary particles.

Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles:

Using the relativistic momentum formula from special relativity

Here is Planck's constant and the relativistic momentum of the electron.

While virtual particles obey conservation of energy and momentum, they can have any energy and momentum, even one that is not allowed by the relativistic energy-momentum relation for the observed mass of that particle.

The specific impulse of relativistic rockets is the same as the effective exhaust velocity, despite the fact that the nonlinear relationship of velocity and momentum as well as the conversion of matter to energy have to be taken into account ; the two effects cancel each other.