Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S27

Newton's first law says that a body retains uniform rectilinear motion, including rest as the zero-velocity case, unless impressed forces change that motion. Its physical content includes the existence of a common class of inertial frames: free bodies do not each require an unrelated frame chosen just to straighten their own trajectory. Uniformity refers to the inertial time parameter, and straightness to the spatial structure of those frames. Everyday slowing under friction is compatible with this claim because it involves an interaction, not an exception to inertia.

Newton's second law relates the impressed resultant force to change of momentum. In modern continuous-force notation,so for constant mass it becomes $m\mathbf{a}={\mathbf{F}}_{\mathrm{net}}$. An impulsive version is $\Delta \mathbf{p}=\int \mathbf{F}dt$. The change is directed along the impressed resultant, not necessarily along the body's current velocity. The original wording concerns change of quantity of motion; the differential equation is its modern continuous-time formulation.^[ref]

These are substantive physical claims only when mass, interactions and the inertial structure are constrained independently. If “force” merely abbreviates whatever equals $m\mathbf{a}$ in any arbitrarily chosen coordinates, the second law by itself predicts nothing. Actual applications combine the law with independently testable interaction laws, vector addition of forces, initial conditions and a specification of the inertial frames.

Showing that acceleration is unchanged by a uniform boost is necessary, but it is not enough to establish the Galilean covariance of Newton's second law. One must specify how time, positions, masses, forces and any background physical systems transform, and show that the actual force laws retain their form.

For a constant spatial rotation $R$, boost velocity $\mathbf{U}$ and translations, writeDifferentiation gives ${\mathbf{v}}^{\prime }=R\mathbf{v}-\mathbf{U}$ and ${\mathbf{a}}^{\prime }=R\mathbf{a}$. Assuming invariant inertial mass,Consequently the equation has the same form if the resultant physical force satisfies ${\mathbf{F}}^{\prime }=R\mathbf{F}$. For a pure boost, this reduces to ${\mathbf{F}}^{\prime }=\mathbf{F}$. More explicitly, if a force is represented by a function of positions, velocities and other physical data, those arguments and data must transform so that its primed value is $R$ times the old value. It is this condition on the complete dynamical model, not the typography of $\mathbf{F}=m\mathbf{a}$, that establishes covariance.

An instantaneous inverse-square interaction depending on the separation of two particles passes this test: their common boost cancels from the separation, and a rotation rotates the force vector. A resistance law $\mathbf{F}=-\zeta (\mathbf{v}-\mathbf{u})$ also passes when the medium's velocity $\mathbf{u}$ transforms with the body. In contrast, keeping $\mathbf{F}=-\zeta \mathbf{v}$ in every boosted frame while silently resetting the same medium to rest changes the physical problem. That model has not been shown covariant by noting ${\mathbf{a}}^{\prime }=\mathbf{a}$.

Likewise, the full Maxwell equations are not made Galilean covariant merely because some equation of particle motion is written as force equals mass times acceleration. The covariance of a framework has to be checked together with its interaction laws. Newton's own Corollary V expresses the corresponding equivalence of uniformly translating, nonrotating systems for the mechanics under consideration.^[ref]

Within an already specified inertial frame, with constant mass, substituting zero resultant force into Newton's second law does give zero acceleration. In that limited mathematical sense, force-free uniform motion is a special case. The reason for retaining Newton's first law is that its physical role is broader than this substitution.

The second law does not, just by its algebra, identify the coordinate systems in which acceleration measures the response to physical interactions. A free body described in coordinates $x^{\prime }=x-\frac{1}{2}A{t}^2$ has ${\ddot{x}}^{\prime }=-A$, despite experiencing no impressed interaction. If the second law is restricted to inertial frames, the first law supplies a criterion for that restriction and asserts that a common inertial structure exists. If instead one introduces a fictitious force $-mA$ to preserve the equation in the accelerated coordinates, that coordinate-dependent bookkeeping should not be confused with a newly discovered physical interaction.

Nor can the first law simply be used as a circular definition in which “free” means “unaccelerated” and “inertial” means “whatever makes the chosen body free”. The framework requires empirical identification of interactions and a mutually consistent class of free-body trajectories. Removing an influence or improving isolation should make the same inertial description work for many bodies, not merely one fitted trajectory.

Thus the two laws have distinguishable jobs: the first articulates the force-free inertial background; the second quantifies departures caused by impressed forces. A modern axiomatization may build that background into its spacetime structure and then omit a separately numbered first law. That is a legitimate reorganization, but it incorporates the first law's content rather than obtaining all of it from the equation $\mathbf{F}=m\mathbf{a}$. Newton states the laws separately in the opening axioms.^[ref]

Einstein's distinctive achievement in 1905 was a reorganization of the physical interpretation and foundations of the theory, not the invention of every transformation formula or every previously known optical effect. He took the equivalence of inertial descriptions and a light-propagation postulate as principles from which to construct a general kinematics of rods, clocks and electromagnetic processes, without assigning explanatory work to an observable state of rest of the ether.^[ref]

The nineteenth-century comparison matters. Hendrik Antoon Lorentz's 1895 theory explicitly describes matter moving through a stationary ether and introduces transformed quantities to relate moving and stationary descriptions. Such work supplied genuine physical insight and important mathematical antecedents; it should not be dismissed as an absence of relativity reasoning.^[ref] His 1904 paper, a useful immediate pre-Einstein comparison rather than a nineteenth-century publication, begins from electron theory and adds assumptions about moving material systems. It already seeks a systematic explanation of the failure to detect motion, rather than an unrelated hypothesis for every experiment.^[ref]

The conceptual change can be seen by asking what a moving clock's “local time” represents. In an ether-based explanation there may remain a privileged time relative to which the moving apparatus is dynamically distorted. In Einstein's formulation the times assigned by identically constructed, appropriately synchronized clocks in each inertial system enter the physical description on the same footing. One should not infer a physically privileged inertial system simply from calling one set of coordinates stationary.

This also changes the scope of the explanation. A special electron model might account for the behaviour of one class of instruments. A relativity principle constrains any acceptable dynamics of measuring devices and matter: if one kind of clock revealed absolute uniform motion, the principle would fail. The result is a framework within which more detailed constructive theories can be developed, rather than a substitute for all such theories.

A careful assessment therefore credits both continuity and innovation. Transformation mathematics and contraction hypotheses had antecedents; the operational treatment of distant time, the equal standing of inertial descriptions, and the use of general principles to unify kinematics and electrodynamics form the characteristic 1905 synthesis. It does not follow that an empirically indistinguishable preferred-frame ontology is logically impossible. What is removed is its necessity in this physical formulation. Nor should the separate 1905 mass–energy paper be silently treated as a section of the June relativity paper.

Yes. Differential ageing between twins who separate and reunite can be calculated entirely within special relativity, provided gravity is negligible and ideal clock behaviour is specified. The theory is not confined to particles that remain inertial throughout their histories: accelerated timelike trajectories can be described in a single inertial coordinate system in flat Minkowski spacetime.

The invariant quantity recorded by an ideal clock is its proper time,Take one twin at rest in an inertial laboratory. Let the other travel out at speed $v$ for laboratory time $T/2$ and return at the same speed for $T/2$, idealizing the turnaround as very short. On reunion,Both times are compared at the same reunion event, so this is not a disagreement about observing a distant clock or choosing a convention for distant synchronization. Einstein's 1905 discussion already considers clock transport along a closed path; the proper-time integral supplies the general modern calculation.^[ref]

The apparent paradox applies reciprocal inertial time dilation as if both twins used one inertial frame from departure to reunion. They do not. The traveller changes inertial frames at the turn, and the associated assignment of simultaneity to distant events changes with it. Reciprocal time dilation for each separate inertial leg is compatible with unequal total proper times along the two worldlines.

Acceleration marks the asymmetry of this itinerary, but is not a universal extra term that locally slows a clock by an amount proportional to its acceleration. Smooth the turnaround and integrate along the resulting worldline: the answer depends on the full path. In the short-turnaround limit the extra proper time during the turn becomes small while the differential ageing of the long legs remains finite. One can even compare successive inertial travellers who relay clocks or records, avoiding the mistaken inference that general relativity is needed to describe the inertial legs.

The clock hypothesis adds the physical idealization that a suitable clock records metric proper time without additional acceleration-dependent damage or internal disturbance. A real biological twin or laboratory clock can of course suffer such disturbances; those are questions about the device's dynamics, not a logical contradiction in the relativistic prediction. General relativity is needed if significant spacetime curvature or gravity is part of the experiment, not merely because a worldline bends in a flat spacetime diagram.

Bell's warning concerns the order of explanation. A formal change of coordinates can be mistaken for a physical mechanism, or for a proof that ordinary questions about material rods and clocks have become meaningless. In the cited essay he recommends first following a concrete model's equations of motion, so that changes in a moving system's shape and period are understood before a general correspondence between frames is used. The phrase quoted in the question occurs in note 10, following the discussion of a classical charged-particle model.^[ref]

A useful independent diagnostic is to ask whether a proposed account predicts an actual stress. Consider two ships with identical acceleration schedules in one inertial laboratory and a fragile thread between them. Their laboratory separation remains fixed. Once both coast at speed $v$, the separation of the attachment points measured in their final common rest frame is $\gamma L$, where $L$ was the fixed laboratory separation. A thread whose unstressed rest length is $L$ must stretch to span those points. For a sufficiently fragile thread it breaks. Saying merely that all contraction is a matter of description misses a mechanical consequence of the specified history. Conversely, length contraction is not an additional universal force pulling every object shorter regardless of its constraints: the stress response depends on its material dynamics and acceleration programme.

The same methodological point applies to clocks. Their ability to realize the relativistic time standard requires compatible physical laws. Calling a variable a new time coordinate does not by itself explain why an atomic transition, an oscillator and a material measuring instrument behave consistently with that variable. Lorentz-covariant dynamics can connect these descriptions without introducing a second set of observable clock readings.

Bell's lesson need not be inflated into a rejection of Einstein's principles or of Minkowski geometry. His comparison expressly recognizes the power of the principle approach and does not require adopting an empirically preferred ether frame. The defensible caution is that covariance, observable physical behaviour and an ontology of spacetime are different claims. Geometrical explanations are useful when their connection to matter and measurement is understood; premature philosophising begins when a coordinate transformation is made to settle those further issues without argument.

There is an important structural difference, but the issue needs two distinctions. The freedom to change distant clock labels is not the same as the absence of a physically preferred simultaneity relation; and an inability to measure a one-way signal speed without synchronization is not by itself a proof that all proposed temporal structures are mere conventions.

In a standard Einstein synchronisation procedure, a clock at A sends a light signal at $t_1$, receives its return at $t_3$, and assigns the reflection event at B the time $(t_1+t_3)/2$. More generally one could assignFor a fixed spatial distance $d$ with round-trip duration $2d/c$, the two assigned one-way speeds become $c/(2ϵ)$ and $c/[2(1-ϵ)]$. Their reciprocal travel times still add to $2d/c$. Thus the two-way measurement alone does not establish the midpoint rule. Einstein presents synchronization as a definition supported by consistency conditions, rather than as a direct comparison of already synchronized distant clocks.^[ref]

Both Newtonian mechanics and special relativity allow coordinated relabellings such as $t_{\kappa }=t+\kappa x/c$. They can be used as nonstandard coordinates if the equations and interpretations of the transformed quantities are changed consistently. A mere relabelling cannot change a reunion of clocks or another local coincidence. Conversely, insisting that the familiar equations must retain their usual simple component form while changing only clock labels is not a neutral test of conventionality.

The difference is in the invariant structures of the standard theories. Newtonian spacetime includes an absolute time, whose equal-time slices are preserved by Galilean transformations. It therefore contains one simultaneity structure shared by its inertial observers. Its instantaneous interaction laws, where adopted, are formulated relative to that time. Ordinary relativistic spacetime contains no analogous observer-independent foliation. Einstein synchronisation defines simultaneity relative to an inertial observer; different inertial observers generally assign different distant events to the same time. This difference survives the freedom to rewrite either theory in awkward coordinates.

It does not follow that any arbitrary synchronization is equally natural relative to the full structure of special relativity. An inertial observer together with the Minkowski metric picks out the usual orthogonal simultaneity hyperplanes, and ideal slow clock transport supplies a physical comparison under assumptions about clock behaviour. A conventionalist must say which such structures and operational assumptions are being held fixed. Likewise, a Newtonian theorist must not confuse positing absolute time with having a theory-independent operational measurement of it.

The best answer is therefore qualified: both theories admit synchronization conventions at the level of coordinate assignment, but their standard physical accounts privilege simultaneity differently. Newtonian absolute simultaneity and the relativistic absence of a shared inertial foliation are genuine theoretical differences. Whether an alternative clock assignment is merely a convention or a rival physical claim depends on whether it only relabels the same structure or adds, removes or changes physical structure.