History of Topics in Special Relativity/Stress-energy tensor (matter)

Overview
The Stress–energy_tensor is defined as


 * $$(a)\ 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}$$

where $$w$$ is the energy density, $$S$$ the energy flux density vector, and $$G$$ the stress tensor. While it was historically introduced in electrodynamics, it was quickly adapted to mechanics in general. A well known example is the perfect fluid stress energy tensor:


 * $$(b)\ T^{\alpha\beta}=\left(\mu_{0}+\frac{p}{c^{2}}\right)u^{\alpha}u^{\beta}+pg^{\alpha\beta}$$

where $$\mu$$ is the mass–energy density, $$p$$ is the hydrostatic pressure, $$u^{\alpha}$$ is the fluid's four velocity, and $$g^{\alpha\beta}$$ is the reciprocal of the metric tensor. In the case of vanishing pressure it becomes the dust solution:


 * $$(c)\ T^{\alpha\beta}=\mu_{0}u^{\alpha}u^{\beta}$$

Its divergence represents the four-force density in that fluid, providing an alternative way to formulate equations of motion.

Tensor (a) was applied to mechanics by, ,. Case (c) was implicitly given by, and explicitly by , #Abraham (1912), , , and in generally covariant form by. Case (b) was given by and in generally covariant form by #Einstein (1914).

Minkowski (1907)
In an appendix to his lecture from December 1907 (published 1908), Hermann Minkowski extended the postulate of relativity to mechanics, defining the space-time vector of second kind:


 * $$S=\left|\begin{array}{cccc}

S_{11}, & S_{12}, & S_{13}, & S_{14}\\ S_{21}, & S_{22}, & S_{23}, & S_{24}\\ S_{31}, & S_{32}, & S_{33}, & S_{34}\\ S_{41}, & S_{42}, & S_{43}, & S_{44} \end{array}\right|=\left|\begin{array}{rrrr} X_{x}, & Y_{x}, & Z_{x}, & -iT_{x}\\ X_{y}, & Y_{y}, & Z_{y}, & -iT_{y}\\ X_{z}, & Y_{z}, & Z_{z}, & -iT_{z}\\ -iX_{t}, & -iY_{t}, & -iZ_{t}, & T_{t} \end{array}\right|$$

which represents the general scheme of mechanical stress energy tensor (a). Subsequently he derived the following relation using four-velocity $$w$$ and rest mass density $$\nu$$ defined as constant:


 * $$\frac{\partial\nu\,w_{h}w_{1}}{\partial x_{1}}+\frac{\partial\nu\,w_{h}w_{2}}{\partial x_{2}}+\frac{\partial\nu\,w_{h}w_{3}}{\partial x_{3}}+\frac{\partial\nu\,w_{h}w_{4}}{\partial x_{4}}$$

Even though Minkowski didn't explicitly mention it, it includes the dust tensor equivalent to (c), from which he derived the mechanical equations of motion.

Abraham (1909-12)
In 1909, Max Abraham pointed out that the relativity principle requires that the mechanical forces must transform like the electromagnetic ones, so there must be a four-dimensional tensor for mechanics (i.e. mechanical stress energy tensor) in analogy to the electromagnetic one, and that the relation $$c^{2}\mathfrak{g}=\mathfrak{S}$$ can alternatively be interpreted as relation between mechanical momentum and energy density first indicated by Planck (1907):


 * $$\begin{matrix}X_{x},\ Y_{y},\ Z_{z};\ X_{y}=Y_{x},\ Y_{z}=Z_{y},\ Z_{x}=X_{z}\\

X_{u}=U_{x}=-i\mathfrak{s}_{x},\ Y_{u}=U_{y}=-i\mathfrak{s}_{y},\ Z_{u}=U_{z}=-i\mathfrak{s}_{z},\ U_{u}=\psi\\ \left[u=ict,\ \mathfrak{s}=\mathfrak{S}/c=c\mathfrak{g}\right] \end{matrix}$$

equivalent to (a).

In 1912, Abraham introduced the expression "world tensor of motion" $$T^{\ast}$$ while formulating his first theory of gravitation. It has ten components representing kinetic stresses, energy flux $$\mathfrak{S}^{\ast}$$ and momentum $$\mathfrak{g}^{\ast}$$ of matter in terms of rest mass density $$\nu$$:


 * $$\begin{matrix}X_{x}^{\ast}=-\nu\dot{x}^{2},\ Y_{y}^{\ast}=-\nu\dot{y}^{2},\ Z_{z}^{\ast}=-\nu\dot{z}^{2}\\

X_{y}^{\ast}=Y_{x}^{\ast}=-\nu\dot{x}\dot{y},\ Y_{z}^{\ast}=Z_{y}^{\ast}=-\nu\dot{y}\dot{z},\ Z_{x}^{\ast}=X_{z}^{\ast}=-\nu\dot{z}\dot{x}\\ \mathfrak{S}_{x}^{\ast}=c^{2}\mathfrak{g}_{x}^{\ast}=icX_{u}^{\ast}=icU_{x}^{\ast}=-ic\nu\dot{x}\dot{u}=\nu c^{2}x\dot{t}\\ \mathfrak{S}_{y}^{\ast}=c^{2}\mathfrak{g}_{y}^{\ast}=icY_{u}^{\ast}=icU_{y}^{\ast}=-ic\nu\dot{y}\dot{u}=\nu c^{2}y\dot{t}\\ \mathfrak{S}_{z}^{\ast}=c^{2}\mathfrak{g}_{z}^{\ast}=icZ_{u}^{\ast}=icU_{z}^{\ast}=-ic\nu\dot{z}\dot{u}=\nu c^{2}z\dot{t}\\ \varepsilon^{\ast}=U_{u}^{\ast}=-\nu\dot{u}^{2}=\nu c^{2}\dot{t}^{2}=\mu c^{2}\varkappa^{-1}\\ \left[\mu=\nu\dot{t}=\nu\varkappa^{-1},\ \mathfrak{S}^{\ast}=c^{2}\mathfrak{g}^{\ast}=\mu c^{2}\varkappa^{-1}\cdot\mathfrak{v}=\varepsilon^{\ast}\cdot\mathfrak{v}\right] \end{matrix}$$

equivalent to (c). Then he combined $$T^{\ast}$$ with the world tensor $$T$$ (representing the electromagnetic-, gravitational-, and stress field) in order to formulate the momentum and energy conservation theorems.

Nordström (1910–13)
Gunnar Nordström (1910) explicitly formulated a “four-dimensional tensor” consisting of rest mass density $$\gamma$$ and four-velocity $$\mathfrak{a}$$


 * $$\gamma\mathfrak{a}_{m}\mathfrak{a}_{n}$$

equivalent to (c). Nordström used this tensor to formulate the four-force density based on the assumption of variable rest mass density: and alternatively based on the assumption of constant rest mass density equivalent to (c).

In 1913, he wrote the above tensor in the form


 * $$\nu\mathfrak{B}_{x}\mathfrak{B}_{y},\dots$$

equivalent to (c) and called it "material tensor", then he combined it with the “elastic stress tensor” p in order to reformulated Laue's symmetrical four dimensional tensor T representing spatial stresses and mechanical momentum and energy density:


 * $$\begin{align}T_{xx} & =p_{xx}+\nu\mathfrak{B}_{x}^{2}\\

& \cdots\\ T_{uu} & =p_{uu}+\nu\mathfrak{B}_{u}^{2}\\ T_{xy} & =p_{xy}+\nu\mathfrak{B}_{x}\mathfrak{B}_{y}\\ & \cdots\\ & \cdots\\ T_{zu} & =p_{zu}+\nu\mathfrak{B}_{z}\mathfrak{B}_{u} \end{align} $$

which can be used to add an elastic component $$\mathfrak{K}^{e}$$ to the four-force-density $$\mathfrak{K}$$ to give equation of motion. He went on to employ this notion in his theory of gravitation.

Laue (1911-20)
In the first textbook on relativity (1911), Max von Laue recognized that the "world tensor" $$T$$ (i.e. stress-energy tensor) not only applies to electrodynamics but to mechanics as well, and that any form of ponderomotive force $$F$$ must be based on such a world tensor, implying the complete reduction of mechanical inertia to energy and stresses. It includes the relation between momentum density and energy flux $$\mathfrak{g}=\tfrac{\mathfrak{S}}{c^{2}}$$ (“inertia of energy” according to Planck), and the Lorentz transformation of the tensor components into rest frame $$K^0$$:


 * $$\begin{matrix}T=\begin{matrix}\mathbf{p}_{xx} & \mathbf{p}_{xy} & \mathbf{p}_{xz} & \frac{i}{c}\mathfrak{S}_{x}\\

\mathbf{p}_{yx} & \mathbf{p}_{yy} & \mathbf{p}_{yz} & \frac{i}{c}\mathfrak{S}_{y}\\ \mathbf{p}_{zx} & \mathbf{p}_{zy} & \mathbf{p}_{zz} & \frac{i}{c}\mathfrak{S}_{z}\\ \frac{i}{c}\mathfrak{S}_{x} & \frac{i}{c}\mathfrak{S}_{y} & \frac{i}{c}\mathfrak{S}_{z} & -W \end{matrix}\\ \hline \mathfrak{S}_{x}=\frac{qc^{2}}{c^{2}-q^{2}}\left(\mathbf{p}_{xx}^{0}+W^{0}\right),\ \mathfrak{S}_{y}=\frac{qc}{c^{2}-q^{2}}\mathbf{p}_{xy}^{0},\ \mathfrak{S}_{z}=\frac{qc}{c^{2}-q^{2}}\mathbf{p}_{xz}^{0}\\ W=\frac{c^{2}W^{0}+q^{2}\mathbf{p}_{xx}^{0}}{c^{2}-q^{2}},\ \mathbf{p}_{yy}=\mathbf{p}_{yy}^{0},\ \mathbf{p}_{zz}=\mathbf{p}_{zz}^{0},\ \mathbf{p}_{yz}=\mathbf{p}_{yz}^{0}\\ \mathbf{p}_{xx}=\frac{c^{2}\mathbf{p}_{xx}^{0}+q^{2}W^{0}}{c^{2}-q^{2}},\ \mathbf{p}_{xy}=\frac{c\mathbf{p}_{xy}^{0}}{\sqrt{c^{2}-q^{2}}},\ \mathbf{p}_{xz}=\frac{c\mathbf{p}_{xz}^{0}}{\sqrt{c^{2}-q^{2}}}\\ \hline W=\frac{c^{2}}{c^{2}-q^{2}}\left\{ W^{0}+\frac{1}{c^{2}}\left(\mathfrak{q}\left[\mathfrak{q}\mathbf{p}^{0}\right]\right)\right\} \\ \mathfrak{g}=\frac{1}{c^{2}}\mathfrak{S}=\frac{\mathfrak{q}}{c^{2}-q^{2}}\left\{ W^{0}+\frac{1}{q^{2}}\left(\mathfrak{q}\left[\mathfrak{q}\mathbf{p}^{0}\right]\right)\right\} +\frac{1}{c\sqrt{c^{2}-q^{2}}}\left\{ \left[\mathfrak{q}\mathbf{p}^{0}\right]-\frac{\mathfrak{q}}{q^{2}}\left(\mathfrak{q}\left[\mathfrak{q}\mathbf{p}^{0}\right]\right)\right\} \\ \hline \mathfrak{S}_{x}\left(1+q^{2}/c^{2}\right)=q\left(W+\mathbf{p}_{xx}\right),\ \mathfrak{S}_{y}=q\mathbf{p}_{xy},\ \mathfrak{S}_{z}=q\mathbf{p}_{xz}\\ \mathfrak{S}+\mathfrak{q}\frac{(\mathfrak{q}\mathfrak{S})}{c^{2}}=W\mathfrak{q}+[\mathfrak{q}\mathbf{p}]=c^{2}\mathfrak{g}+\mathfrak{q(qg)}\\ \hline E=\frac{c}{\sqrt{c^{2}-q^{2}}}\left\{ E^{0}+\frac{1}{c^{2}}\left(\mathfrak{q}\left[\mathfrak{q},\ \int\mathbf{p}^{0}dV^{0}\right]\right)\right\} \\ \mathfrak{G}=\frac{\mathfrak{q}}{c\sqrt{c^{2}-q^{2}}}\left\{ E^{0}+\frac{1}{q^{2}}\left(\mathfrak{q}\left[\mathfrak{q},\ \int\mathbf{p}^{0}dV^{0}\right]\right)\right\} +\frac{1}{c^{2}}\left\{ \left[\mathfrak{q},\ \int\mathbf{p}^{0}dV^{0}\right]-\frac{\mathfrak{q}}{q^{2}}\left(\mathfrak{q}\left[\mathfrak{q},\ \int\mathbf{p}^{0}dV^{0}\right]\right)\right\} \\ \left[E=\int WdV=\frac{\sqrt{c^{2}-q^{2}}}{c}\int WdV^{0},\ \mathfrak{G}=\int\mathfrak{g}dV=\frac{\sqrt{c^{2}-q^{2}}}{c}\int\mathfrak{g}dV^{0}\right] \end{matrix}$$

equivalent to (a). From that he derived the total static system in which the total pressure vanishes


 * $$\begin{matrix}\int\mathbf{p}^{0}dV^{0}=0\\

E=\frac{c}{\sqrt{c^{2}-q^{2}}}E^{0},\ \mathfrak{G}=\frac{\mathfrak{q}}{c\sqrt{c^{2}-q^{2}}}E^{0}\\ \left[E=\int W\,dS,\ \mathfrak{G}=\frac{1}{c^{2}}\int\mathfrak{S}\,dS\right] \end{matrix}$$

In the second edition (1912, published 1913) he slightly rewrote the above mechanical world tensor and its Lorentz transformation in terms of momentum density $$\mathfrak{g}$$ as


 * $$\begin{matrix}T=\begin{matrix}\mathbf{p}_{xx} & \mathbf{p}_{xy} & \mathbf{p}_{xz} & ic\mathfrak{g}_{x}\\

\mathbf{p}_{yx} & \mathbf{p}_{yy} & \mathbf{p}_{yz} & ic\mathfrak{g}_{y}\\ \mathbf{p}_{zx} & \mathbf{p}_{zy} & \mathbf{p}_{zz} & ic\mathfrak{g}_{z}\\ \frac{i}{c}\mathfrak{S}_{x} & \frac{i}{c}\mathfrak{S}_{y} & \frac{i}{c}\mathfrak{S}_{z} & -W \end{matrix}\quad\mathfrak{g}=\tfrac{\mathfrak{S}}{c^{2}}\\ \hline \mathfrak{g}_{x}=\frac{\mathfrak{q}_{x}\left(\mathbf{p}_{xx}^{0}+W^{0}\right)}{c^{2}-q^{2}},\ \mathfrak{g}_{y}=\frac{\mathfrak{q}_{x}\mathbf{p}_{xy}^{0}}{c^{2}-q^{2}},\ \mathfrak{g}_{z}=\frac{\mathfrak{q}_{x}\mathbf{p}_{xz}^{0}}{c^{2}-q^{2}}\\ W=\frac{c^{2}W^{0}+q^{2}\mathbf{p}_{xx}^{0}}{c^{2}-q^{2}},\ \mathbf{p}_{yy}=\mathbf{p}_{yy}^{0},\ \mathbf{p}_{zz}=\mathbf{p}_{zz}^{0},\ \mathbf{p}_{yz}=\mathbf{p}_{yz}^{0}\\ \mathbf{p}_{xx}=\frac{c^{2}\mathbf{p}_{xx}^{0}+q^{2}W^{0}}{c^{2}-q^{2}},\ \mathbf{p}_{xy}=\frac{c\mathbf{p}_{xy}^{0}}{\sqrt{c^{2}-q^{2}}},\ \mathbf{p}_{xz}=\frac{c\mathbf{p}_{xz}^{0}}{\sqrt{c^{2}-q^{2}}}\\ \hline W=\frac{c^{2}}{c^{2}-q^{2}}\left\{ W^{0}+\frac{1}{c^{2}}\left(\mathfrak{q}\left[\mathfrak{q}\mathbf{p}^{0}\right]\right)\right\} \\ \mathfrak{g}=\frac{\mathfrak{q}}{c^{2}-q^{2}}\left\{ W^{0}+\frac{1}{q^{2}}\left(\mathfrak{q}\left[\mathfrak{q}\mathbf{p}^{0}\right]\right)\right\} +\frac{1}{c\sqrt{c^{2}-q^{2}}}\left\{ \left[\mathfrak{q}\mathbf{p}^{0}\right]-\frac{\mathfrak{q}}{q^{2}}\left(\mathfrak{q}\left[\mathfrak{q}\mathbf{p}^{0}\right]\right)\right\} \\ \hline E=\frac{c}{\sqrt{c^{2}-q^{2}}}\left\{ E^{0}+\frac{1}{c^{2}}\left(\mathfrak{q}\left[\mathfrak{q},\ \int\mathbf{p}^{0}dV^{0}\right]\right)\right\} \\ \mathfrak{G}=\frac{\mathfrak{q}}{c\sqrt{c^{2}-q^{2}}}\left\{ E^{0}+\frac{1}{q^{2}}\left(\mathfrak{q}\left[\mathfrak{q},\ \int\mathbf{p}^{0}dV^{0}\right]\right)\right\} +\frac{1}{c^{2}}\left\{ \left[\mathfrak{q},\ \int\mathbf{p}^{0}dV^{0}\right]-\frac{\mathfrak{q}}{q^{2}}\left(\mathfrak{q}\left[\mathfrak{q},\ \int\mathbf{p}^{0}dV^{0}\right]\right)\right\} \end{matrix}$$

equivalent to (a).

In the fourth edition of his book (1921), he defined the following tensor in relation to the dynamics of mass points in terms of rest energy density $$W^{0}$$, rest energy $$E^{0}$$, rest volume $$V^{0}$$:


 * $$\begin{matrix}T_{jk}=W^{0}Y_{j}Y_{k}\\

\int T_{ik}\delta V^{0}=E^{0}Y_{i}Y_{k} \end{matrix}$$

equivalent to the dust tensor (c).

Herglotz (1911)
Gustav Herglotz gave a complete theory of elasticity in special relativity which he defined using coordinates $$x\dots$$ after deformation, $$\xi\dots$$ and $$\xi^{0}\dots$$ before deformation, from which he derived the deformation quantities $$a_{ij}$$ and $$A_{ij}$$, together with the kinetic potential $$\varPhi$$. He defined the Euler equations of motion using stress-energy tensor $$F_{ij}$$, whose components can be related to momentum density $$\mathfrak{X},\mathfrak{Y},\mathfrak{Z}$$, energy density $$\mathfrak{E}$$, velocity u,v,w, as well as "relative" stresses $$S_{ij}$$:


 * $$\begin{matrix}\varDelta F_{ij}=\overline{\Omega}_{ij}+\frac{1}{2}\frac{\Omega}{A_{44}}\frac{\partial A_{44}}{a_{i4}}a_{j4}\\

\hline \mathfrak{X}=F_{14},\ \mathfrak{Y}=F_{24},\ \mathfrak{Z}=F_{34},\ \mathfrak{E}=-F_{44}\\ F_{23}=F_{32},\ F_{31}=F_{13},\ F_{12}=F_{21}\\ F_{14}+F_{41}=0,\ F_{24}+F_{42}=0,\ F_{34}+F_{43}=0\\ \hline \begin{align}S_{11} & =F_{11}+u\mathfrak{X}, & S_{12} & =F_{12}+v\mathfrak{X}, & S_{13} & =F_{13}+w\mathfrak{X},\\ S_{21} & =F_{21}+u\mathfrak{Y}, & S_{22} & =F_{22}+v\mathfrak{Y}, & S_{23} & =F_{23}+w\mathfrak{Y},\\ S_{31} & =F_{31}+u\mathfrak{Z}, & S_{32} & =F_{32}+v\mathfrak{Z}, & S_{33} & =F_{33}+w\mathfrak{Z}, \end{align} \end{matrix}$$

and showed how to modify the above components $$\mathfrak{X,Y,Z,E},S_{ij}$$ using mass density m and pressure p, so as to become the hydrodynamic stress-energy tensor:


 * $$\begin{matrix}F_{ij}=p\delta_{ij}-\frac{m}{a_{44}^{2}}\frac{\partial A_{44}}{\partial a_{i4}}a_{j4}\\

\hline \mathfrak{X}=mu,\ \mathfrak{Y}=mv,\ \mathfrak{Z}=mw,\ \mathfrak{E}=m-p\\ S_{11}=S_{22}=S_{33}=p\\ S_{23}=S_{32}=S_{31}=S_{13}=S_{12}=S_{21}=0\\ \left[m=\frac{F+p}{1-s^{2}},\ s=\sqrt{u^{2}+v^{2}+w^{2}}\right] \end{matrix}$$

which corresponds to (b) or in case of vanishing pressure to (c). He consequently derived the equations of motion and four-force density (X,Y,Z,T).

Lewis/Wilson (1912)
Edwin Bidwell Wilson and Gilbert Newton Lewis (1912) devised an alternative 4D vector calculus based on Dyadics which, however, never gained widespread support. They defined the dyadic $$\mu_{0}\mathbf{ww}$$ using four-velocity $$\mathbf{w}$$ and rest mass density $$\mu_{0}$$ in order to formulate the fundamental equation of hydrodynamics:


 * $$\begin{matrix}\mu_{0}\mathbf{ww}\\

\lozenge\cdot(\mu_{0}\mathbf{ww})=0\\ \left[\lozenge=\mathbf{k}_{1}\frac{\partial}{\partial x_{1}}+\mathbf{k}_{2}\frac{\partial}{\partial x_{2}}+\mathbf{k}_{3}\frac{\partial}{\partial x_{3}}-\mathbf{k}_{4}\frac{\partial}{\partial x_{4}}\right] \end{matrix}$$

equivalent to (b,c).

Einstein (1913-16)
In 1913, in the context of his Entwurf theory (a precursor of general relativity), Albert Einstein discussed the motion of continuously distributed incoherent masses in gravitational fields, by using the contravariant "stress energy tensor of the material flow"


 * $$\Theta_{\mu\nu}=\varrho_{0}\frac{dx_{\mu}}{ds}\cdot\frac{dx_{\nu}}{ds}$$

equivalent to (c) in the case of $$g_{\mu\nu}$$ being the Minkowski tensor. In a remark to that paper, they used the tensor


 * $$\mathfrak{T}_{\sigma\nu}=\sum_{\mu}\sqrt{-g}g_{\sigma\mu}\Theta_{\mu\nu}$$

which Einstein in 1914 denoted as "energy tensor of the ponderable mass flow":


 * $$\mathfrak{T}_{\sigma}^{\nu}=\rho_{0}\sqrt{-g}\frac{dx_{\nu}}{ds}\sum_{\mu}g_{\sigma\mu}\frac{dx_{\mu}}{ds}$$

and by including pressure p, the tensor of an ideal fluid becomes in case of adiabatic motion


 * $$\mathfrak{T}_{\sigma}^{\nu}=-p\delta_{\sigma}^{\nu}\sqrt{-g}+\rho_{0}\sqrt{-g}(1+p+P)\frac{dx_{\nu}}{ds}\sum_{\mu}g_{\sigma\mu}\frac{dx_{\mu}}{ds}$$

which Einstein in 1916 wrote as


 * $$\left.T_{\sigma}\right.^{\alpha}=-\left.\delta_{\sigma}\right.^{\alpha}p+g_{\sigma\beta}\frac{dx_{\beta}}{ds}\frac{dx_{\alpha}}{ds}\varrho$$ (with $$\varrho\Rightarrow\varrho-p$$ in special relativity).

equivalent to (b) in the case of $$g_{\mu\nu}$$ being the Minkowski tensor.

Kottler (1914)
Friedrich Kottler credits for introducing the "Nordström tensor" in terms of rest mass density $$\nu_{0}$$:


 * $$T^{(hk)}=\nu_{0}\frac{dx^{(h)}}{d\tau}\frac{dx^{(k)}}{d\tau}=\nu\frac{dx^{(h)}}{dt}\frac{dx^{(k)}}{dt}$$

equivalent to (c), from which he also derived the equations of motion. He commented that this tensor is realized when Laue's (1911) total static system is averaged over the total volume.