6 Mass, Energy and Momentum

Figure 8 depicts a moving object with spatial velocity $\mathbf{V}$ of magnitude $v$ , as measured by an observer at point L, at rest.

Figure 8: 4D velocity of magnitude $c$ in $x'_4$ of an object at L. An observer at rest at L has velocity of magnitude $c$ in $x_4$ .

The vector sum of spatial and time-velocities reflects the four-velocities of the observer (along $x_4$ ) and the moving object (along $x'_4$ ). It follows naturally that the Lorentz invariant $m_0 c$ ($m_0$ is the rest mass) in the moving object A can be decomposed in

$$m_0^2 c^2=m_0^2 \chi^2+m_0^2 v^2$$ (24)

which, using the identities $E=\gamma m_0 c^2$ and $p=\gamma m_0 v$ , is equivalent to the classical equation

$$E^2/c^2=m_0^2 c^2+p^2$$ (25)

$E$ being the total energy and $p$ being the spatial momentum.

The components in the right part of Eq. (24) cannot simply be interpreted as, respectively, the object's momenta in the time dimension and the spatial dimension of the rest frame of the observer. There is an obvious problem in the fact that the factor $\gamma$ is involved in the expressions for $E$ and $p$ . If we multiply the factor $\gamma ^2$ into all three elements of Eq. (24) we get:

$$\gamma ^2m_0^2 c^2=\gamma ^2m_0^2 \chi^2+\gamma ^2m_0^2 v^2$$ (26)

which describes triangle LK'M (if $m_0$ is set to 1). This alternative form for Eq. (24) immediately shows the meaning of its components. They now correspond one to one with the components in Eq. (25): $\gamma m_0 c=E/c$ , $\gamma m_0 \chi=m_0c$ , $\gamma m_0 v=p$ . The factor $\gamma m_0 c$ is however not invariant under rotations in SO(4), while $m_0 c$ is. [Note that although $m_0 c$ is indeed Lorentz invariant from the perspective of the observer, its physical meaning in its own rest frame is the moving object's time-momentum. The same invariant value can be found in the rest frame of the observer (see also Fig. 9) but should then be read as $\gamma m_0\chi$ .] The Lagrangian formalism for this situation has been worked out by Montanus in [2]. The reader is therefore referred to this source for the detailed derivation. The generic principles used for such 5D situations (or more generally 4D with the addition of an extra parameter to keep track of the progress of the object along its world-line) appear in Goldstein [8]. The latter however uses the classical indefinite Minkowski metric as a basis for the development of the relativistic Lagrangian $\Lambda$ where Montanus uses a positive definite metric like in this article. A short overview of the main equations is given here.

In agreement with classical mechanics it is assumed that the variation according to Hamilton's principle:

$$\delta I=\int_{x_{5_{(1)}}}^{x_{5_{(2)}}} \Lambda(x_\mu, u_\mu)dx_5$$ (27)

is an extremum, where $u_\mu=dx_\mu/dx_5$ . The corresponding Euler-Lagrange equations of motion are:

$$\frac{\partial \Lambda}{\partial x_\mu}-\frac{d}{dx_5}(\partial \Lambda/\partial u_\mu)=0$$ (28)

leading to a possible relativistic Lagrangian for a free object in the absence of a forcefield (so the potential energy equals zero):

$$\Lambda=m_0u_\mu u^\mu$$ (29)

which equals, as a result of the universal velocity magnitude $c$ for the free particle in 4D space-time:

$$\Lambda=m_0c^2$$ (30)

The latter is to be interpreted as the 'kinetic energy' of the particle in four dimensions, which is a fundamentally different concept than kinetic energy in three dimensions. It corresponds to the total energy of a particle at rest. Other solutions for $\Lambda$ are possible but the essential element is that any solution is a constant in 4D space-time.

The relativistic Lagrangian $\Lambda$ shows that the factor $\gamma$ in Eq. (26) must be a result of our confinement to a 3D subspace of 4D space-time. In order to maintain conservation laws for energy and momentum, while only being able to measure their 'projections' to our 3D space, the factor $\gamma$ is an artificial necessity. It vanishes for a hypothetical observer with full 4D observational skills, who measures the object's speed and energy as constants.