Energy-momentum tensor

The energy-momentum tensor is a tensor that is mainly used in field theory . It can be stated and interpreted in the following general form:

${\displaystyle (T^{\alpha \beta })={\begin{pmatrix}w&{\frac {S_{x}}{c}}&{\frac {S_{y}}{c}}&{\frac {S_{z}}{c}}\\{\frac {S_{x}}{c}}&G_{xx}&G_{xy}&G_{xz}\\{\frac {S_{y}}{c}}&G_{yx}&G_{yy}&G_{yz}\\{\frac {S_{z}}{c}}&G_{zx}&G_{zy}&G_{zz}\end{pmatrix}}}$

• ${\displaystyle w}$is an energy density (energy per volume ). At low speeds it is dominated by the density of the mass , but photons that have no mass also contribute to the energy density with their energy . ${\displaystyle E=h\cdot \nu }$
• ${\displaystyle (S_{x},S_{y},S_{z})}$is an energy flux density (energy density multiplied by a speed ).
• ${\displaystyle c}$is the speed of light in a vacuum.
• ${\displaystyle G_{ik}}$is the negative of Maxwell's stress tensor in the case of application to electromagnetic radiation . It includes the spatial impulse transport, e.g. B. in the diagonal terms the pressure that the electromagnetic radiation field can exert. The off-diagonal terms of this stress tensor describe shear stresses .

Within the framework of the special theory of relativity and the general theory of relativity , the energy - momentum tensor is a second order four-tensor .

Geometric spatiotemporal interpretation in 4D language

For the sake of simplicity, Planck units are used in this article . The speed of light is normalized to one, so that mass and energy are identified with one another based on the equivalence of mass and energy . ${\displaystyle c}$ ${\displaystyle E=mc^{2}}$ ${\displaystyle m}$ ${\displaystyle E}$

• The component ( energy density , mass density ) describes the energy flow (mass flow) in a time-like direction, i.e. the energy flow through a space-like 3D volume element. ${\displaystyle T^{00}}$
• The components ; (spatial energy flow, spatial mass flow) describe the energy flow density (mass flow density) in the spatial i direction, i.e. the energy flow through a 3D volume element with one time-like and two space-like axes. ${\displaystyle T^{i0}}$ ${\displaystyle i=1,\dotsc ,3}$
• The components ; (Momentum density) describe the momentum flow of the k th component of the momentum in time-like direction, i.e. the momentum flow of the k th component of the momentum through a space-like 3D volume element. ${\displaystyle T^{0k}}$ ${\displaystyle k=1,\dotsc ,3}$
• The components ; ( Momentum current density ) describe the momentum flow of the k th component of the momentum in the spatial i direction, i.e. the momentum flow of the k th component through a 3D volume element with one time-like and two space-like axes. ${\displaystyle T^{ik}}$ ${\displaystyle i,k=1,\dotsc ,3}$

The symmetry contains the following information: ${\displaystyle T^{\alpha \beta }=T^{\beta \alpha }}$

• ${\displaystyle T^{\alpha 0}=T^{0\alpha }}$: The mass flow density (energy flow density) is equal to the momentum density; this is a consequence of the focal point .
• The shear stresses are symmetrical: A transport of the k th component of the momentum in the i direction is always accompanied by an equally large transport of the i th component of the momentum in the k direction ( ); this is a consequence of the conservation of angular momentum . ${\displaystyle i,k=1,\dotsc ,3}$

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

${\displaystyle \nabla _{\alpha }T^{\alpha \beta }=0}$

described, where denotes the energy-momentum tensor of all fields involved . Describes only the energy-momentum tensor of a field that interacts with other fields, for example electromagnetic radiation alone (see below), then the energy-momentum balance equation reads ${\displaystyle (T^{\alpha \beta })}$ ${\displaystyle (T^{\alpha \beta })}$

${\displaystyle \nabla _{\alpha }T^{\alpha \beta }=f^{\beta }}$,

where the right side denotes the four force density, i.e. the four momentum exchange with other fields per 4D volume element. The components with describe the momentum balance, the component with the energy balance (mass balance). ${\displaystyle \beta =1,\dotsc ,3}$ ${\displaystyle \beta =0}$

Together with a suitable volume shape , the energy-momentum tensor can be used to calculate the energy-momentum four-vector that belongs to this 3D volume element.

The energy-momentum tensor of electrodynamics

In the Heaviside-Lorentz system of units

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

${\displaystyle (T^{\alpha \beta })={\begin{pmatrix}{\frac {1}{2}}(E^{2}+B^{2})&({\vec {E}}\times {\vec {B}})^{T}\\{\vec {E}}\times {\vec {B}}&{\frac {1}{2}}(E^{2}+B^{2})\delta _{ik}-E_{i}E_{k}-B_{i}B_{k}\end{pmatrix}}}$

(In the Gaussian system of units the representation differs from the one given here by the factor .) ${\displaystyle {\frac {1}{4\pi }}}$

• ${\displaystyle E}$is the symbol for the electric field strength .
• ${\displaystyle B}$is the symbol for the magnetic flux density .
• ${\displaystyle \delta _{ik}}$denotes the Kronecker Delta .

• The component of the tensor is the energy density of the electromagnetic field. ${\displaystyle T_{00}}$
• ${\displaystyle {\vec {S}}={\vec {E}}\times {\vec {B}}}$is called the Poynting vector . It describes the energy flux density and the momentum density of the electromagnetic field.
• The components , describe the negative of the stress tensor (pulse current density) of the electromagnetic field, that in the diagonal elements of the (radiation) in pressure and the non-diagonal components of the shear stress of the field. ${\displaystyle {\tfrac {1}{2}}(E^{2}+B^{2})\delta _{ik}-E_{i}E_{k}-B_{i}B_{k}}$ ${\displaystyle i,k=1,2,3}$

The stress-energy tensor is a - matrix , as is a vector with three components. ${\displaystyle (T^{\alpha \beta })}$ ${\displaystyle 4\times 4}$ ${\displaystyle {\vec {E}}\times {\vec {B}}}$

In the SI system of units

The energy-momentum tensor looks like this in SI units :

${\displaystyle (T^{\alpha \beta })={\begin{pmatrix}{\tfrac {1}{2}}(\varepsilon _{0}E^{2}+{\frac {B^{2}}{\mu _{0}}})&c\varepsilon _{0}({\vec {E}}\times {\vec {B}})^{T}\\c\varepsilon _{0}{\vec {E}}\times {\vec {B}}&{\tfrac {1}{2}}(\varepsilon _{0}E^{2}+{\frac {B^{2}}{\mu _{0}}})\delta _{ik}-\varepsilon _{0}E_{i}E_{k}-{\frac {1}{\mu _{0}}}B_{i}B_{k}\end{pmatrix}}}$

• ${\displaystyle \varepsilon _{0}}$is the symbol for the electric field constant .
• ${\displaystyle \mu _{0}}$is the symbol for the magnetic field constant .

The Poynting vector now has the following form:

${\displaystyle {\vec {S}}=c^{2}\varepsilon _{0}{\vec {E}}\times {\vec {B}}}$

The conversion from the representation in the International System of Units (SI) to the simpler Heaviside-Lorentz system of units with the convention is done simply by omitting the constants , and . ${\displaystyle c=1}$ ${\displaystyle \varepsilon _{0}}$ ${\displaystyle \mu _{0}}$ ${\displaystyle c}$

The Maxwell stress tensor is contained in the energy-momentum tensor with a negative sign. In SI units, Maxwell's stress tensor has the form:

${\displaystyle \sigma _{ij}=\epsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2}}{\bigl (}{\epsilon _{0}E^{2}+{\tfrac {1}{\mu _{0}}}B^{2}}{\bigr )}\delta _{ij}}$

Relativistic 4D notation for the electromagnetic energy-momentum tensor

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

${\displaystyle T^{\alpha \beta }=F^{\alpha \gamma }F_{\gamma }^{\;\;\beta }-{\frac {1}{4}}g^{\alpha \beta }F_{\mu \nu }F^{\nu \mu }}$.

Notations used:

• ${\displaystyle (F_{\alpha \beta })={\begin{pmatrix}0&E_{1}&E_{2}&E_{3}\\-E_{1}&0&-B_{3}&B_{2}\\-E_{2}&B_{3}&0&-B_{1}\\-E_{3}&-B_{2}&B_{1}&0\end{pmatrix}}}$denotes the electromagnetic field strength tensor ( ) and ${\displaystyle c=1}$
• ${\displaystyle g=\operatorname {diag} (1,-1,-1,-1)}$denotes the metric tensor of the special theory of relativity. This tensor is used to pull the indices up and down .

Balance equations for the energy-momentum tensor in electrodynamics

In 3D notation

Hereinafter referred to

• ${\displaystyle {\vec {S}}={\vec {E}}\times {\vec {H}}}$ the Poynting vector,
• ${\displaystyle \rho }$ the electrical charge density of a charged matter field,
• ${\displaystyle {\vec {j}}}$ the electrical current density of a charged matter field.

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

${\displaystyle {\frac {\partial }{\partial t}}\left[{\tfrac {1}{2}}(E^{2}+B^{2})\right]+\operatorname {div} {\vec {S}}={\vec {j}}\cdot {\vec {E}}}$

The left side shows the local energy balance of the electromagnetic field, the right side the power density of the electromagnetic field in the matter field. This relationship is also known as Poynting's theorem.

${\displaystyle {\frac {\partial }{\partial t}}S_{k}+{\frac {\partial }{\partial x_{i}}}\left[{\tfrac {1}{2}}(E^{2}+B^{2})\delta _{ik}-E_{i}E_{k}-B_{i}B_{k}\right]=({\vec {j}}\times {\vec {B}}+\rho {\vec {E}})_{k}\quad k=1,\dotsc ,3}$

The left side represents the local momentum balance of the electromagnetic field, the right side the Lorentz force density of the electromagnetic field on the charged matter field.

In 4D notation

In special relativistic 4D notation, these two balance equations can also be summarized as follows:

${\displaystyle {\frac {\partial }{\partial x^{\alpha }}}T_{\;\;\beta }^{\alpha }=j^{\alpha }F_{\alpha \beta }\quad \beta =0,\dotsc 3}$

Here denotes the four-vector of the electromagnetic four-current. ${\displaystyle (j^{\alpha })=(\rho ,{\vec {j}})}$

The right side again gets the interpretation of a Lorentzian four force density (four momentum transfer per 4D volume element). ${\displaystyle j^{\alpha }F_{\alpha \beta }}$

The energy-momentum tensor in general relativity

The energy-momentum tensor of matter and radiation forms the right-hand side of Einstein's field equations of general relativity and thus acts as a "source term" for the curvature of space-time . What is new compared to Newton's theory of gravity is that all components of the tensor play the role of “sources” of gravity, not just the mass density . At moderate pressures, shear stresses and speeds in laboratory experiments, you practically do not notice this, because measured in natural units, the mass density of matter is usually many orders of magnitude greater than all other components of the energy-momentum tensor. ${\displaystyle T^{00}}$

The energy-momentum tensor of hydrodynamics

The energy-momentum tensor of hydrodynamics goes into Einstein's field equations and enables the specification of solutions to the differential equations with which the dynamics of the cosmos can be described. In theoretical physics textbooks containing chapters on cosmology , it is usually given in contravariant representation as follows:

${\displaystyle T^{\alpha \beta }=\left(\rho +{\frac {P}{c^{2}}}\right)u^{\alpha }u^{\beta }-P\;g^{\alpha \beta }}$

• ${\displaystyle (u^{\alpha })}$is the speed of four .
• ${\displaystyle P}$ describes the isotropic pressure in a local inertial system of a freely falling observer.
• ${\displaystyle \rho }$ is the mass density in a local inertial system.
• ${\displaystyle (g^{\alpha \beta })}$ is the metric tensor of general relativity.
• ${\displaystyle c}$ is the magnitude of the vacuum speed of light.

This description of the energy-momentum-tensor applies to a set of liquid or gas particles that may be called an ideal gas or an ideal liquid. It is therefore assumed that the pressure in the rest frame of each particle is isotropic . Thermal conduction and viscosity are also neglected and therefore cannot be described using this representation of the energy-momentum tensor.

In cosmology, galaxies are viewed as elements of an ideal cosmic fluid. The galaxy does not expand due to its own gravity. However, due to the cosmic expansion , it is moving away from all other galaxies. An observer who moves with this galaxy is considered to be at rest relative to it. In this sense, the galaxy forms the rest system of the observer moving with it. In such a rest system, the vector reduces to the speed of four of the galaxy . This rest system is at the same time the system of a freely falling observer. One can therefore find coordinates so that in this system the metric tensor of the special theory of relativity can be used instead of the general metric tensor . ${\displaystyle (u^{\alpha })=(c,0,0,0)}$ ${\displaystyle (g^{\alpha \beta })}$ ${\displaystyle (\eta ^{\alpha \beta })}$

This simplifies the representation of the energy-momentum tensor:

${\displaystyle (T^{\alpha \beta })={\begin{pmatrix}\rho c^{2}&0&0&0\\0&P&0&0\\0&0&P&0\\0&0&0&P\end{pmatrix}}}$

If the pressure also disappears , the energy-momentum tensor consists only of the energy density ( ): ${\displaystyle P}$ ${\displaystyle e=\rho c^{2}}$

${\displaystyle (T^{\alpha \beta })={\begin{pmatrix}\rho c^{2}&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}}}$

In general, however, this representation is only valid for one point in space-time. For larger areas of spacetime, the general metric tensor of spacetime must be used.

Impenergie

In their book Physics of Spacetime , John Archibald Wheeler and Edwin F. Taylor proposed the “second great essential unit” - in addition to spacetime , unified space and time in a uniform four-dimensional structure - the momentum as a spatial component and the momentum as a temporal component Contains energy, to be designated with the term Impenergie .

literature

• Richard Feynman : Lectures on Physics Volume 3: Quantum Mechanics. Oldenbourg 1991 (SI), ISBN 3-486-25134-1 .
• Walter Greiner : Classical Electrodynamics. Verlag Harri Deutsch, 1991 (Gauss system), ISBN 3-8171-1184-3 .
• Torsten Fließbach : General Theory of Relativity. BI Wissenschaftsverlag, 1990, ISBN 3-8274-1356-7 (with a section on hydrodynamics and a chapter on cosmology).
• Edwin F. Taylor , John Archibald Wheeler : Physics of Space-Time . Spectrum, 1994, ISBN 3-86025-123-6 .

Web links

• Introductory presentation slides on the electromagnetic energy pulse tensor (PDF; 948 kB).

credentials

1. ^ Charles W. Misner, Kip S. Thorne, John Archibald Wheeler: Gravitation . WH Freeman, San Francisco September 1973, ISBN 0-7167-0344-0 . , Chapter 5.2 "Three-Dimensional Volumes and Definition of the Stress Energy Tensor", p. 130 f.
2. ^ M. Alcubierre, "Introduction to 3 + 1 Numerical Relativity", point 1.12, page 32, 2008