Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S27

The age difference is the difference between proper times along two worldlines joining the same departure and reunion events. In the inertial Earth frame a traveler who spends total coordinate time $T$ moving at speed magnitude $v$, with negligible-duration turnaround, accumulates $T\sqrt{1-v^2/c^2}$ while the stay-at-home clock accumulates $T$. These are invariant readings at reunion. Acceleration permits the change of traveling inertial frame, but the result is determined by the whole path, not a universal rule that acceleration itself slows clocks.

Relativity of simultaneity is important when explaining the same result using the traveler's successive inertial coordinates. At the turnaround, the traveler changes which distant Earth event is assigned the same time. The simultaneity slice changes, producing a jump in the coordinate time attributed to Earth in the ideal instantaneous limit. On each long inertial leg the traveler can assign a slow rate to the distant Earth clock without contradicting Earth's larger total elapsed time, because those assignments do not cover the reunion comparison with a single simultaneity convention.

The key distinction is between explaining a coordinate-based apparent symmetry and deriving an invariant result. One can calculate both proper-time integrals in one inertial frame without ever using the traveler's simultaneity jump. Conversely, claiming that each twin must simply see the other younger confuses two reciprocal comparisons between different sets of simultaneous distant events. Radar signals and Doppler shifts provide an operational account with finite signal travel times and the same reunion readings.

Thus relativity of simultaneity resolves a specific misleading inference in the paradox, but is not an extra physical mechanism that changes a clock at a distance. The worldline asymmetry and the invariant proper-time calculation suffice for the prediction; simultaneity explains why reciprocal inertial time-dilation statements cannot be pasted together indiscriminately.^[ref]

Special relativity supplies a tenseless mathematical representation: events have spacetime relations, proper-time intervals and causal order whether or not a coordinate labels one event “now.” For spacelike separated events, different inertial frames disagree about simultaneity and temporal order. The invariant structure therefore does not select a global present consisting of all and only events simultaneous with an observer's current event.

That supports an argument for eternalism if one adds that what exists should not depend on an arbitrary inertial frame and that no extra preferred foliation is available. On those assumptions, restricting existence to a frame-dependent simultaneity plane looks unjustified. A spacetime account can instead represent an observer's temporal location as it represents spatial location, without introducing an objective moving present.

The conclusion is not a formal derivation of metaphysics from the Lorentz transformations alone. Relativity establishes the metric and its symmetries, not a complete semantics for existence, becoming or experience. A presentist could add a preferred but empirically hidden slicing, though this adds structure unsupported by ordinary relativistic predictions. Local becoming along a worldline or causal partial orders are other possibilities, and do not reinstate a unique invariant global simultaneity relation.

A tenseless description also does not erase every temporal distinction. Timelike-separated events retain invariant order, light cones distinguish causal possibilities, and clocks measure nonzero proper durations. The absence of a global privileged “now” does not identify past and future dynamically or explain the thermodynamic arrow. Relativity strongly constrains a global present and makes a block representation natural; whether that representation exhausts temporal reality requires additional philosophical premises.

A coordinate transformation considered solely as a relabeling of events has no independent observable outcome: changing labels changes neither a detector record nor a clock comparison. But the Lorentz transformations also express substantive relations between operationally realized inertial coordinates. With clocks and rulers specified, their velocity-dependent time dilation, length relations and velocity composition make testable claims about how material systems and electromagnetic signals behave.

The distinction matters for synchronization. One-way coordinate light speed depends on how distant clocks are synchronized; a conventional alteration changes the displayed transformation law while leaving two-way timing observations unchanged. It would therefore be incorrect to regard every coefficient in an unspecified coordinate formula as a separately measurable physical fact. Testable content comes from the transformation together with the protocol and dynamical behavior of the systems implementing the coordinates.

Tests can compare clock proper times, decay lifetimes or resonance behavior across relative motion and orientation. Agreement supports Lorentz-invariant kinematics and the associated matter dynamics over the range tested; it does not prove exact invariance at every possible scale. A rival theory with an undetectable preferred frame can reproduce the same observations by changing its explanatory commitments. Empirical equivalence of that rival does not make the observable clock relations untestable.

Thus the answer is qualified: transformations as passive mathematical changes are not experiments, while Lorentzian relations among specified measuring procedures have substantial empirical consequences. Separating these roles avoids both treating coordinate convention as a measured force and declaring the entire relativistic structure immune to evidence.

Homogeneity makes the inertial transformation linear. Align origins at $t=t^{\prime }=0$, let the second origin have velocity $v$, and impose isotropy and reciprocity. Then $x^{\prime }=\gamma (v)(x-vt)$, $y^{\prime }=y$, $z^{\prime }=z$. If the same limiting signal speed $c$ has trajectories $x=\pm ct$ in both systems, substituting these two trajectories fixes $t^{\prime }=\gamma (v)(t-vx/c^2)$. Requiring the inverse to have the same form with $-v$ fixes ${\gamma }^2(1-v^2/c^2)=1$. Continuity at rest selects the positive square root:Light enters this derivation through the empirical/in-principle operational identification of the invariant speed and Einstein synchronization. Group, homogeneity and isotropy arguments can instead derive a family with a universal parameter $K$, $t^{\prime }=\gamma (t-Kvx)$ and $\gamma =(1-K{v}^2{)}^{-1/2}$. They do not by themselves determine whether $K=0$ or a positive finite value. Empirical light propagation or other relativistic dynamics identifies $K=1/c^2$ for our world. Consequently a derivation need not assume light as its first premise, but it must acquire the physical invariant-speed structure somewhere rather than extract it from linearity alone.^[ref]

Newton's first law says that an unforced body maintains uniform straight-line motion in an inertial frame. It can function partly as a criterion for identifying inertial frames, but that does not automatically empty it of empirical content. The substantive claim is that there is a common class of frames in which independently isolated bodies have this behavior, rather than needing a different accelerated frame for each trajectory.

Testing requires auxiliary judgments about forces and measurement. If every unexpected acceleration is redescribed as an unknown force, or if a new coordinate system is tailored to every body, the law becomes insulated by construction. In practice forces are investigated independently through interactions, shielding, environmental changes and comparison among bodies. Reducing drag and varying external influences allow extrapolation toward the ideal. Mutually consistent force-free trajectories and the success of the broader dynamical system then support the inertial-frame hypothesis.

A simple counter to the claim that any coordinate choice suffices is a rotating laboratory: free bodies generally have velocity-dependent Coriolis accelerations and cannot all be made uniformly rectilinear by calling that rotating frame inertial while retaining the same force account. Conversely, no finite experiment establishes complete isolation or exact uniform motion forever. The law is tested as part of a network of force laws, geometrical assumptions and clock/ruler procedures, and may have local rather than global validity in a gravitational theory.

The first law is therefore neither an independently observed slogan with no definitions nor a mere tautology. Its definition-like role fixes a useful framework, while its applicability and the consistency of that framework across systems are answerable to experience.^[ref]