Past exam of the symmetry and relativity course of the University of Oxford / 2019 / 1

Use coordinates $X^{\mu }=(ct,\mathbf{x})$ and the mostly-plus Minkowski metric ${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,1,1)$. Along a future-directed timelike world line, proper time satisfiesThe four-velocity and four-acceleration areThe Minkowski norm is fixed: $U^{\mu }{U}_{\mu }=-c^2$. Differentiating with respect to proper time, using the constant Minkowski metric, givesIn the instantaneous rest reference frame, $U^{\mu }=(c,0,0,0)$, so the orthogonality says $A^0=0$ there. This construction uses a timelike world line; a massless particle has no rest-frame proper time parameter.

Let $m$ denote the constant rest mass, set $x(0)=0$, and put $a=f_x/m$. The relativistic three-force is $f_x=d{p}_x/dt$, soIt follows thatIntegrating the velocity with the initial position fixed givesThese formulas allow either sign of $a$; their $a\to 0$ limit is rest at the origin. For $a\ne 0$, the trajectory obeysThis is hyperbolic motion. Its coordinate acceleration decreases while its proper acceleration has constant magnitude $|a|$. In particular, a constant three-force does not give constant coordinate acceleration at relativistic speed.

The upper-left panel shows the three causal structure regions, with null boundaries included in the appropriate causal future and causal past. For the charged trajectory, use part b with $a=q{E}_x/m$, choosing the $x$ direction so that $a>0$. The electric field gives the constant three-force $q{E}_x$ and hence the shifted hyperbolaIt starts with a vertical tangent at the origin, stays strictly inside the future light cone, and asymptotes to $x=ct-c^2/a$. The other panels implement the coordinate and topology changes used below; they are original spacetime diagrams, not copies of source artwork.

A future-directed causal curve from the origin has $|dx/dt|\le c$. Integrating this inequality gives the causal futureThe straight timelike or null segment proves sufficiency. This is the blue region above the two null rays in the upper-left spacetime diagram of part c.

Reverse the endpoints of the future-directed causal curve. The origin can be reached precisely from its causal past,This is the orange region below the two null rays in part c. The origin itself belongs to both the causal future and causal past by the usual zero-length-curve convention.

Outside both light cones, the separation from the origin is spacelike. Thus the events in neither the causal past nor the causal future areNo causal curve can connect either endpoint to the other. These are the two green side regions in part c; the strict inequality excludes the causally reachable null boundaries.

For the standard passive Lorentz boost, write $\beta =v/c$ andA proper orthochronous Lorentz transformation preserves the sign of the time component of a nonzero future causal vector and preserves its Minkowski norm. ThereforeThe upper-right panel of part c uses $x^{\prime }$ horizontally and $c{t}^{\prime }$ vertically. Its null rays still have slopes $\pm 1$. The grey dashed old axes satisfy $x^{\prime }=-\beta c{t}^{\prime }$ for the old time axis and $c{t}^{\prime }=-\beta x^{\prime }$ for the old spatial axis; changing the coordinate grid has not changed which events are reachable.

The same Lorentz transformation maps the past light cone to itself. In the primed axes the causal past isthe orange region in the upper-right spacetime diagram of part c. Future and past are not interchanged by the standard boost.

The invariant spacelike condition is $x^{\prime 2}-c^2{t}^{\prime 2}>0$. Hence the primed-coordinate events in neither the causal future nor the causal past obeyThe green regions in part c retain their shape in primed coordinates. A particular spacelike event's coordinate time may change sign, but it never enters either light cone.

For causal reachability on a spatial circle, put $P=2\pi R$ and use a representative $\bar{x}\in [-P/2,P/2)$ of the spatial circle. The lower-left panel of part c is a fundamental strip with its vertical edges identified. On the covering space the charged world line is still the solution of part b; on the quotient it isA jump between opposite strip edges is a coordinate seam, not a discontinuity of the physical world line. At its $n$th complete lap, $x=nP$, soFor $a=q{E}_x/m>0$, the speed increases on each return while remaining below $c$. Its kinetic energy increases byThere is no single-valued electrostatic potential energy $-q{E}_x\bar{x}$ on a circle: the nonzero electromotive force is $\oint \mathbf{E}\cdot d\mathbf{l}=E_{x}P$. A globally defined magnetic vector potential one-form $A_x=-E_{x}t$ instead makes the driving explicit. In an embedded circuit, Faraday's law realizes this electromotive force through changing magnetic flux. More generally an external apparatus maintaining the field supplies the increasing particle energy. In an isolated dynamical field-plus-particle system, backreaction would change the field. Thus this is not a cyclic machine extracting work from a conservative potential, and it does not violate conservation of energy.

A causal curve can wind around the spatial circle. Lift its endpoint to $x+nP$ on the covering space. A lift is reachable from the origin if $ct\ge |x+nP|$ for at least one integer $n$. With shortest circular distancethe causal future isIn the centred fundamental strip, $d=|\bar{x}|$. For $t\ge P/(2c)=\pi R/c$, every spatial point is in the causal future. This produces the fully blue upper part of the lower-left panel in part c.

The causal past follows from the same covering-space argument with reversed endpoints:For $t\le -\pi R/c$, the whole spatial circle can send a causal signal to the origin. The lower-left spacetime diagram in part c is therefore completely orange sufficiently far below the origin.

The complement of the causal future and causal past isThis green region is confined to $|t|<\pi R/c$ and meets the identified strip edges. Spatial periodicity creates alternative signal paths, but time remains noncompact and no closed timelike curve follows from this spatial identification.

For causal reachability with periodic time, let $P_t=2\pi T$. The lower-right panel of part c identifies its top and bottom edges. The vertical curve at fixed $x$ is a closed timelike curve: after a positive proper duration $P_t$, it returns to the same spacetime event. The quotient is a mathematically well-defined flat Lorentzian manifold, but it violates chronology and admits no usual global causal ordering or ordinary global initial-value evolution.

These are physical causality problems, rather than an algebraic contradiction in the metric. Fields and particle histories must satisfy periodic consistency conditions; an arbitrary initial state followed by irreversible evolution will generally fail to return to itself after a temporal lap. The fact that the causal past and causal future overlap everywhere, established below, explains why the ordinary three-way causal partition degenerates.

For any representative event $(t,x)$, choose an integer $n$ large enough that $c(t+n{P}_t)>|x|$. A future-directed timelike segment from the origin to that lift projects to a causal curve on the quotient. ThereforeEven points with large $|x|$ can be reached by allowing sufficiently many temporal windings. The purple lower-right panel in part c denotes simultaneous membership in both the causal future and causal past.

Choose an integer $n$ sufficiently negative that $c(t+n{P}_t)<-|x|$. That lift lies in the ordinary causal past of the origin, and its projected causal curve reaches the origin. HenceFuture-directed curves can connect the origin and an arbitrary event in either order. This property includes the origin's nontrivial closed timelike curves.

Every event is in both the causal future and causal past, soDrawing just the nearest two null rays on a temporally identified strip would miss winding causal curves and give an incorrect nonempty exterior region.