Phenomenological Representation of the Energy Tensor of Matter.

1. Phenomenological Representation of the Energy Tensor of Matter.

Hydrodynamical Equations. We know that matter is built up of electrically charged particles, but we do not know the laws which govern the constitution of these particles. In treating mechanical problems, we are therefore obliged to make use of an inexact description of matter, which corresponds to that of classical mechanics. The density $σ$ , of a material substance and the hydrodynamical pressures are the fundamental concepts upon which such a description is based. Let $σ_{0}$ be the density of matter at a place, estimated with reference to a system of co-ordinates moving with the matter. Then $σ_{0}$ , the density at rest, is an invariant. If we think of the matter in arbitrary motion and neglect the pressures (particles of dust in vacuo, neglecting the size of the particles and the temperature), then the energy tensor will depend only upon the velocity components, $u_{ν}$ and $σ_{0}$ . We secure the tensor character of $T_{μ ν}$ by putting

T_{μ ν} = σ_{0} u_{μ} u_{ν},

(66)

in which the $u_{μ}$ , in the three-dimensional representation, are given by (53). In fact, it follows from (66) that for $q = 0$ , $T_{44} = - σ_{0}$ (equal to the negative energy per unit volume), as it should, according to the theorem of the equivalence of mass and energy, and according to the physical interpretation of the energy tensor given above. If an external force (four-dimensional vector, $K_{μ}$ ) acts upon the matter, by the principles of momentum and energy the equation

K_{μ} = \frac{\partial T_{μ ν}}{\partial x_{ν}}

must hold. We shall now show that this equation leads to the same law of motion of a material particle as that already obtained. Let us imagine the matter to be of infinitely small extent in space, that is, a four-dimensional thread; then by integration over the whole thread with respect to the space co-ordinates

x_{1}

x_{2}

x_{3}

, we obtain

\begin{array}{rcl} \int K_{1} d x_{1} d x_{2} d x_{3} & = & \int \frac{\partial T_{14}}{\partial x_{4}} d x_{1} d x_{2} d x_{3} \\ = & - i \frac{d}{d l} \{\int σ_{0} \frac{d x_{1}}{d τ} \frac{d x_{4}}{d τ} d x_{1} d x_{2} d x_{3}\} . \end{array}

Now $\int d x_{1} d x_{2} d x_{3} d x_{4}$ is an invariant, as is, therefore, also $\int σ_{0} d x_{1} d x_{2} d x_{3} d x_{4}$ . We shall calculate this integral, first with respect to the inertial system which we have chosen, and second, with respect to a system relatively to which the matter has the velocity zero. The integration is to be extended over a filament of the thread for which $σ_{0}$ may be regarded as constant over the whole section. If the space volumes of the filament referred to the two systems are $d V$ and $d V_{0}$ respectively, then we have

\int σ_{0} d V d l = \int σ_{0} d V_{0} d τ

and therefore also

\int σ_{0} d V = \int σ_{0} d V_{0} \frac{d τ}{d l} = \int d m i \frac{d τ}{d x_{4}} .

If we substitute the right-hand side for the left-hand side in the former integral, and put $\frac{d x_{1}}{d τ}$ outside the sign of integration, we obtain,

K_{x} = \frac{d}{d l} (m \frac{d x_{1}}{d τ}) = \frac{d}{d l} (\frac{m q_{x}}{\sqrt{1 - q^{2}}}) .

We see, therefore, that the generalized conception of the energy tensor is in agreement with our former result.

The Eulerian Equations for Perfect Fluids. In order to get nearer to the behaviour of real matter we must add to the energy tensor a term which corresponds to the pressures. The simplest case is that of a perfect fluid in which the pressure is determined by a scalar $p$ . Since the tangential stresses $p_{x y}$ , etc., vanish in this case, the contribution to the energy tensor must be of the form $p δ_{ν μ}$ . We must therefore put

T_{μ ν} = σ u_{μ} u_{ν} + p δ_{μ ν} .

(67)

At rest, the density of the matter, or the energy per unit volume, is in this case, not

σ

but

σ - p

. For

- T_{44} = - σ \frac{d x_{4}}{d τ} \frac{d x_{4}}{d τ} - p δ_{44} = σ - p .

In the absence of any force, we have

\frac{\partial T_{μ ν}}{\partial x_{ν}} = σ u_{ν} \frac{\partial u_{μ}}{\partial x_{ν}} + u_{μ} \frac{\partial (σ u_{ν})}{\partial x_{ν}} + \frac{\partial p}{\partial x_{μ}} = 0 .

If we multiply this equation by

u_{μ}

(= \frac{d x_{μ}}{d τ})

and sum for the

μ

’s we obtain, using (52),

- \frac{\partial (σ u_{ν})}{\partial x_{ν}} + \frac{d p}{d τ} = 0,

(68)

where we have put $\frac{\partial p}{\partial x_{μ}} \frac{d x_{μ}}{d τ} = \frac{d p}{d τ}$ . This is the equation of continuity, which differs from that of classical mechanics by the term $\frac{d p}{d τ}$ , which, practically, is vanishingly small. Observing (68), the conservation principles take the form

σ \frac{d u_{μ}}{d τ} + u_{μ} \frac{d p}{d τ} + \frac{\partial p}{\partial x_{μ}} = 0 .

(69)

The equations for the first three indices evidently correspond to the Eulerian equations. That the equations (68) and (69) correspond, to a first approximation, to the hydrodynamical equations of classical mechanics, is a further confirmation of the generalized energy principle. The density of matter and of energy has the character of a symmetrical tensor.

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