Stress–energy tensor

The energy -momentum tensor is a mathematical object in physics. It is given in its general form as follows:

• Is an energy density (energy per volume)
• Is an energy flow density, the speed of light in vacuum
• Is known as Maxwellian stress tensor, it contains the spatial pulse transport, for example, in the diagonal terms of the pressure, the radiation field can have. The non- diagonal terms of the stress tensor describing shear stresses.

Within the framework of special relativity as the given energy -momentum tensor is a Vierertensor step ( 2.0 ).

The energy density is dominated at low speeds of the density of the mass, but also photons have no rest mass, contribute their energy to the energy density.

An energy flux density is an energy density multiplied by a rate.

Geometric space-time interpretation in 4D speech

For simplicity in this article, Planck units are used. So the speed of light to unity is normalized so be identified with each other due to the equivalence of mass and energy, mass and energy.

• The component ( energy density, mass density) describes the flow of energy ( mass flow) in time-like direction, ie, the flow of energy through a space -like 3-D volume element.
• The components; (spatial energy flow, spatial mass flow ) describe the energy flux density ( mass flux ) in spatial i- direction, ie the energy flow. due to a 3D volume element with a timelike and two spacelike axes
• The components; ( Pulse density ) describe the impulse flow of the k-th component of the impulse in a time direction like, ie, the momentum flux of the k- th component of the pulse by a space -like 3D volume element.
• The components; ( Pulse current density) describe the impulse flow of the k-th component of the pulse in the I- dimensional direction, thus the momentum flux of the k - th component of a 3D volume element with a time-like, and two space -like shafts.

The symmetry contains the following information:

• The mass flux ( energy flux density ) is equal to the pulse density; which is a consequence of the focus set.
• The shear stresses are symmetrical: a transport component of the k-th pulse in the i direction is always accompanied by an equally large transport of the i- th component of the impulse in the direction of k (); which is a consequence of conservation of angular momentum.

The energy-momentum conservation in the theory of relativity is determined by the balance equation

Described when the energy -momentum tensor of all fields involved called. Describes only the energy - momentum tensor of a field which interacts with other fields, for example, the electromagnetic radiation alone (see below), this is the energy-momentum balance equation

With the right side, the four-force density, ie the four-momentum exchange with other fields per 4D volume element respectively. The components describe the momentum balance, the component energy balance ( mass balance ).

Differential Geometric you can interpret the energy -momentum tensor as a vector-valued 3- form: Each 3D volume element is assigned to the energy-momentum four-vector, which flows through this 3D volume element. The conversion into a Vierertensor step ( 0.2 ) is then carried out with the Hodge operator.

The energy -momentum tensor in general relativity theory

The energy -momentum tensor of matter and radiation forms the right side of the Einstein - Hilbert 's field equations and thus acts as a " source term " for the curvature of space-time. New compared to the Newtonian gravitational theory is that all the components of the tensor playing the role of " sources " of gravitation, not only the mass density. At moderate pressures, shear stresses and velocities in laboratory experiments, practically not noticed this because measured in natural units, the mass density of matter is usually many orders of magnitude larger than all other components of the energy - Impulstensors.

The energy -momentum tensor of electrodynamics

In the Lorentz - Heaviside system of units

In the electrodynamics in the Heaviside - Lorentz system of units ( rationalized Cgs ) of the energy -momentum tensor of the electromagnetic field is:

( In the Gaussian system of units, the representation of the given here differs by a factor. )

• Is the symbol for the electric field strength.
• Is the symbol for the magnetic flux density.
• Denotes the Kronecker delta.

• The component of the tensor is the energy density of the electromagnetic field.
• Is called the Poynting vector. It describes the current density and the pulse energy density of the electromagnetic field.
• The components that describe the stress tensor ( pulse current density) of the electromagnetic field, that is, in the diagonal elements of the (radiation ) and pressure in the non-diagonal components of the shear stress of the field.

The energy - momentum tensor is a matrix, as is a vector with three components.

In the SI unit system

The energy -momentum tensor sees in SI units as follows:

• Is the symbol for the electric field constant.
• Is the symbol for the magnetic field constant.

The Poynting vector now has the following form:

Relativistic 4D notation for the electromagnetic energy -momentum tensor

In relativistic 4D notation can describe the energy -momentum tensor of the electromagnetic field as follows:

Notations used:

• Denotes the electromagnetic field strength tensor () and
• Denotes the metric tensor of special relativity. Shifting up and pulling down, the index is with this tensor.

Balance equations for the energy - momentum tensor in electrodynamics

In 3D notation

Hereinafter referred

• The Poynting vector,
• The electric charge density of a charged matter field,
• The electric current density of the charged matter field,

The Maxwell equations for the electromagnetic field imply the following balance equations for the components of the energy - Impulstensors:

The left hand side to check out the local energy balance of the electromagnetic field is, and the right side of the power density of the electromagnetic field on the matter field. This relationship is also known as a set of Poynting.

The left hand side to check out the local balance of momentum of the electromagnetic field is, and the right side of the Lorentz force density of the electromagnetic field on the charged matter field.

In 4D notation

In special- relativistic 4D notation can also combine these two balance equations as follows:

Herein, the four-vector of the electromagnetic four- current.

The right side gets back the interpretation of a Lorentz four-force density between ( four momentum transfer per 4D volume element).

The energy-momentum tensor of the hydrodynamics

The energy-momentum tensor of the hydrodynamics is included in the Einstein's field equations and allows the specification of solutions of the differential equations that the dynamics of the universe can be described.

He is in textbooks of theoretical physics that contain chapters on cosmology, usually specified in contravariant representation as follows:

• Is the four-velocity
• Describes the pressure (such as a radiation field )
• Is the mass density
• Is the metric tensor of special relativity
• C is the magnitude of the vacuum speed of light

This description of the energy-momentum tensor is a lot of liquid particles ahead are made on the following conditions: It is an ideal fluid before and the pressure is isotropic in the rest frame of each particle.

In cosmology, galaxies are considered as elements of a perfect cosmic fluid. The inner expansion of a galaxy is not considered, they removed it due to the cosmic expansion from all other galaxies, an observer moving with this galaxy, is considered relative to it as dormant.

In this sense, a galaxy is the rest frame of a comoving observer. In such a rest frame, the vector of the four-velocity reduced to.

A further simplification results in the rest frame of the observer by the fact that the metric tensor can be replaced by the metric tensor of special relativity.

Consider as an example the free fall of a Fahrstuhles. The co-moving passenger feels weightless. He rests in its co-moving system that moves due to the Earth's gravity.

Thus, the energy -momentum tensor can be simplified:

Disappears, the pressure, there is the energy -momentum tensor only from the energy density ():