Chapter 22: Space, Time & Relativity

In the Scholium to the Principia (1687), Newton distinguished between absoluteand relative space and time. Absolute space “in its own nature, without relation to anything external, remains always similar and immovable.” Absolute time “of itself, and from its own nature, flows equably without relation to anything external.” Relative space and time are our imperfect, observational measures of these absolutes.

“Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration.”— Isaac Newton, Principia Mathematica, Scholium (1687)

Newton’s famous bucket argument was designed to demonstrate the reality of absolute space. Consider a bucket of water suspended by a twisted rope. When the rope is released:

Newton argued that the concavity of the water’s surface cannot be explained by the water’s motion relative to the bucket (since the surface is flat in Stage 1 and concave in Stage 2, despite the relative motion being present in both cases). Nor can it be explained by motion relative to the fixed stars, since in principle the stars could be removed. The only adequate explanation, Newton concluded, is that the water is rotating with respect to absolute space.

The bucket argument is, at its core, an argument about the reality of absolute acceleration. Newton acknowledged that absolute velocity is undetectable (Galilean relativity), but maintained that absolute acceleration produces observable effects (inertial forces) that cannot be explained relationally.

Leibniz rejected Newton’s absolute space through two powerful philosophical arguments, both deployed in his correspondence with Clarke.

The Shift Argument (Principle of Sufficient Reason)

If absolute space existed, God could have created the material universe shifted five miles to the east, or rotated 90 degrees. But this “shifted world” would be observationally indistinguishable from the actual world. By the Principle of Sufficient Reason, God would have no reason to actualise one rather than the other. Therefore, absolute space (which would make these count as different possibilities) cannot exist.

The Identity Argument (Identity of Indiscernibles)

Leibniz’s Principle of the Identity of Indiscernibles (PII) states that if two putative entities share all their properties, they are in fact identical. Two “different” spatial locations that differ only in their position within absolute space — not in any relational or qualitative property — would violate PII. Therefore, spatial points are not genuine entities; space is constituted by the spatial relations among material bodies.

“I hold space to be something merely relative, as time is; ... I hold it to be an order of coexistences, as time is an order of successions.”— Gottfried Wilhelm Leibniz, Third Letter to Clarke (1716)

Leibniz’s arguments are metaphysically powerful, but the relationist faces a serious technical challenge: how to account for inertial effects (the concavity of the water, the centrifugal force in a rotating frame) without reference to absolute space. This challenge persisted until Mach and Einstein offered new approaches.

Ernst Mach (1883) offered a radical relationist response to the bucket argument. The concavity of the water’s surface is not due to rotation relative to absolute space but to rotation relative to thedistant matter of the universe — the fixed stars and galaxies. If the bucket were at rest and the entire universe rotated around it, Mach conjectured, the same inertial effects would be observed.

This idea — that the local inertial frame is determined by the distribution of matter in the universe as a whole — came to be known as Mach’s principle, a term coined by Einstein himself. Mach’s principle was one of Einstein’s primary motivations in developing general relativity. He hoped that his theory would vindicate Mach by making the inertial structure of spacetime fully determined by the matter content.

The relationship between general relativity and Mach’s principle is, however, complicated. General relativity does allow the matter distribution to influence the inertial structure (the phenomenon offrame-dragging, confirmed experimentally by Gravity Probe B in 2011). But it also admits vacuum solutions — spacetimes with inertial structure but no matter at all — which violate Mach’s principle. Whether general relativity is Machian remains a disputed question.

Einstein’s special theory of relativity (1905) overthrew Newton’s absolute time by showing thatsimultaneity is relative to the observer’s state of motion. Two events that are simultaneous for one inertial observer are not simultaneous for another moving relative to the first. The spacetime interval between events, however, is invariant:

This invariant interval defines Minkowski spacetime, the geometric arena of special relativity. The philosophical implications are profound:

• ◆The block universe: If simultaneity is relative, there is no objective “now” that divides past from future. This has led many philosophers to endorse eternalism (the “block universe” view), according to which past, present, and future are equally real.
• ◆Presentism under threat: Presentism, the view that only the present exists, faces severe difficulties in relativistic physics, since there is no frame-independent way to identify “the present.”
• ◆Conventionality of simultaneity: Reichenbach and Grünbaum argued that even within a single frame, the choice of a simultaneity relation involves a conventional element, since it depends on the assumption that the one-way speed of light is isotropic.

The Twin Paradox

The twin paradox — in which one twin travels at high speed and returns to find the other has aged more — is not a genuine paradox but a dramatic illustration of relativistic time dilation. The travelling twin’s worldline through spacetime is longer in spatial extent but shorter in proper time:

Philosophically, the twin paradox illustrates that time is not an absolute parameter but a quantity that depends on the worldline through spacetime. This is sometimes cited as evidence for the reality of spacetime as a four-dimensional entity.

General relativity (1915) is Einstein’s theory of gravitation. Its central insight is that gravity is not a force acting through space but a manifestation of the curvature of spacetime itself. Matter and energy tell spacetime how to curve; curved spacetime tells matter how to move. This is captured by Einstein’s field equations:

Here $G_{\mu\nu}$ is the Einstein tensor (encoding spacetime curvature), $T_{\mu\nu}$ is the stress-energy tensor (encoding matter and energy), $\Lambda$ is the cosmological constant, and $g_{\mu\nu}$ is the metric tensor.

The philosophical significance of general relativity for the substantivalist-relationist debate is complex. On one hand, the metric field $g_{\mu\nu}$ seems to be a genuine physical entity — it has degrees of freedom, carries energy (gravitational waves), and exists even in vacuum. This supports a form of substantivalism. On the other hand, the theory is generally covariant: its laws take the same form in any coordinate system, suggesting that spacetime points have no intrinsic identity.

“Space-time does not claim existence on its own but only as a structural quality of the [gravitational] field.”— Albert Einstein, “Relativity and the Problem of Space” (1952)

The hole argument, revived by John Earman and John Norton (1987) from Einstein’s own unpublished reasoning, poses a dilemma for manifold substantivalism — the view that the points of the spacetime manifold are genuine entities with intrinsic identity.

The argument exploits the general covariance (diffeomorphism invariance) of general relativity. Given any solution $\langle M, g_{\mu\nu}, T_{\mu\nu} \rangle$ to Einstein’s field equations, we can generate a new solution by applying a diffeomorphism (a smooth point-shuffling of the manifold) that is the identity outside a “hole” region $\mathcal{H}$ but reshuffles points inside it. The new solution $\langle M, d^*g_{\mu\nu}, d^*T_{\mu\nu} \rangle$ agrees with the original everywhere outside the hole but differs inside it.

If the manifold substantivalist insists that the points of $M$ have intrinsic identity, then these are two physically distinct solutions that agree on all observable quantities outside the hole. This means that general relativity, interpreted substantivally, is radically indeterministic: the physics outside the hole does not determine the physics inside it.

The standard responses include:

• ◆Sophisticated substantivalism: Spacetime points are real, but their identity is determined by the metric field, not independently. Diffeomorphic models represent the same physical situation.
• ◆Relationism: Spacetime points are not genuine entities. Only the relational structure encoded in the metric matters.
• ◆Structural realism: What is real is the structure — the pattern of relations — not the individual relata. The spacetime points are mere placeholders in a structure.

The hole argument remains one of the most discussed topics in the philosophy of physics. It connects questions about spacetime ontology to broader issues in the philosophy of symmetry, individuality, and structural realism.

The fundamental laws of physics (with minor exceptions involving the weak nuclear force) aretime-reversal invariant: for every process allowed by the laws, the time-reversed process is equally allowed. Yet our experience is saturated with temporal asymmetry: eggs break but do not unbreak; ice melts but does not spontaneously freeze; we remember the past but not the future. How can temporally symmetric laws give rise to a temporally asymmetric world?

The standard answer appeals to the second law of thermodynamics: entropy tends to increase. Ludwig Boltzmann explained this statistically: high-entropy macrostates correspond to vastly more microstates than low-entropy ones, so the evolution toward higher entropy is overwhelmingly probable:

But this immediately raises a puzzle: the same statistical reasoning implies that entropy should also have been higher in the past, which is empirically false. The resolution requires aboundary condition: the universe began in an extraordinarily low-entropy state (the “Past Hypothesis,” as David Albert calls it). The arrow of time is grounded not in the laws of physics but in the initial conditions of the universe.

“The second law of thermodynamics is, I think, the most important law in all of physics — more important even than quantum mechanics or general relativity.”— Sean Carroll, From Eternity to Here (2010)

The philosophical questions are deep: Is the Past Hypothesis a law or a contingent fact? Why was the early universe in such a special state? Can the arrow of time be explained without boundary conditions? These questions connect the philosophy of time to cosmology, statistical mechanics, and the foundations of probability.