Past exam of the relativity course of the University of Cambridge / 2018 / Section B / B2

The two postulates of special relativity are: all nongravitational laws have the same form in every inertial frame, and light in vacuum has the same finite speed of light $c$ in every inertial frame, independently of the motion of its source. The first is Einstein's principle of relativity; the second supplies an experimentally identified invariant speed.

For the stated standard configuration, the Lorentz transformation isTake differentials along the particle's worldline, rather than comparing coordinate displacements at equal times in the two frames. Writing $u_i=d{x}^i/dt$, one obtainsDividing by the transformed time interval proves the velocity-addition formula:For a massive particle, $|\mathbf{u}|<c$ and $|v|<c$, so the denominator $1-v{u}_x/c^2$ is positive.

Let $\mathbf{a}=d\mathbf{u}/dt$ denote coordinate acceleration and use $A^{\mu }$ for four-acceleration, to distinguish these two objects. Applying the chain rule to the Lorentz factor givesSince $dt/d\tau ={\gamma }_u$, differentiation of the four-velocity $U^{\mu }={\gamma }_u(c,\mathbf{u})$ with respect to proper time yieldsThis is also ${\gamma }_u^2[{\gamma }_u^2(\mathbf{u}\cdot \mathbf{a})/c,\mathbf{a}+{\gamma }_u^2(\mathbf{u}\cdot \mathbf{a})\mathbf{u}/c^2]$, as required.

The invariant Minkowski norm of the four-velocity is $U^{\mu }{U}_{\mu }=c^2$. Differentiating this identity gives $U^{\mu }{A}_{\mu }=0$. In the instantaneous rest frame, $U^{\mu }=(c,0)$, so $A^0=0$ and the spatial components equal the physically experienced proper acceleration $\mathbf{\alpha }$. Therefore ${\alpha }^2=-A^{\mu }{A}_{\mu }$ in the $+---$ metric signature.

For a direct calculation, put $q=\mathbf{u}\cdot \mathbf{a}$ and expand the temporal and spatial squares:The last step uses $1-u^2/c^2={\gamma }_u^{-2}$. This proper acceleration from coordinate velocity and acceleration also reads ${\alpha }^2={\gamma }_u^6{a}_{\parallel }^2+{\gamma }_u^4{a}_{\perp }^2$, providing the familiar one-dimensional limit $\alpha ={\gamma }_u^3|a_{\parallel }|$. Each expression has units ${\mathrm{m}}^2{\mathrm{s}}^{-4}$.

Write ${\Gamma }_a(v)=(1-v^2/a^2{)}^{-1/2}$. The first proposed Lorentz transformation isFor the next transformation, write ${\Gamma }_b(u)=(1-u^2/b^2{)}^{-1/2}$ and substitute the first pair into $x^{\prime \prime }={\Gamma }_b(u)(x^{\prime }-u{t}^{\prime })$ and $t^{\prime \prime }={\Gamma }_b(u)(t^{\prime }-u{x}^{\prime }/b^2)$. The result isThe direct transformation between $S$ and $S^{\prime \prime }$ must belong to the same standard boost family with the same calibration of space and time units. Such a transformation has one common factor multiplying the coefficient of $x$ in $x^{\prime \prime }$ and that of $t$ in $t^{\prime \prime }$. Equality of these coefficients for generic $uv\ne 0$ requiresThe case $uv=0$ alone cannot constrain the scales, but a transformation law must work for arbitrary successive boosts. Once the scales agree, their composed velocity is $V=(u+v)/(1+uv/a^2)$ and the common factor is ${\Gamma }_a(V)$. This establishes the universality of an invariant speed under boost composition within the proposed family.

Finiteness needs physical input. Sending $a^2$ to infinity gives the consistent Galilean transformation $x^{\prime }=x-vt$, $t^{\prime }=t$. Its boosts compose with $V=u+v$ and satisfy a relativity principle. Thus the composition calculation itself does not prove that $a^2$ cannot be infinite. The finite propagation speed of light in vacuum and its invariance, supplied by the second of the postulates of special relativity, rule out this limit for physical spacetime. Under a Galilean transformation, a ray with laboratory speed $c$ would have speed $c-v$ in the moving frame, in conflict with that postulate.

Finally, the common invariant scale must be identified with the physical speed of light, rather than an arbitrary speed measured in one frame. Requiring a ray $x=ct$ also to obey $x^{\prime }=c{t}^{\prime }$ givesHence the physical scale is $a=c$, taking the positive speed. The postulate about light identifies and requires a finite invariant scale; closure alone only establishes its universality. This distinction is necessary for a logically valid answer to the question's request to exclude infinity.