Absolute Space and Absolute Time

Hang a bucket of water from a twisted cord, let it go, and watch. At first the bucket spins and the water sits still, flat. Slowly friction drags the water round, and as it catches up the surface hollows out, climbing the walls. Now stop the bucket: the water keeps turning and the surface stays hollow. Newton put this experiment in the Principia of 1687 and drew a conclusion that has unsettled physicists ever since. The concavity does not track the water's motion relative to the bucket — that was greatest when the surface was flat. It tracks the water's rotation relative to something else, and Newton called that something absolute space. The Scholium to the Definitions says it plainly. "Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external"; absolute space "in its own nature, without relation to anything external, remains always similar and immovable." Two fixed containers — a rigid Euclidean stage and a universal clock — against which all real motion is finally referred. The strange thing is how shy this stage is. Newton himself proved that no mechanical experiment reveals your velocity through absolute space: $\mathbf{F} = m\mathbf{a}$ is identical in every uniformly-moving laboratory, so absolute rest is unfindable. Acceleration is another matter. Spin the bucket and the water tells you; whirl two globes on a cord in an empty universe and the cord pulls taut. Absolute space refuses to show itself to velocity and cannot hide from acceleration — that asymmetry is the real content of this article. Leibniz, Berkeley and, most sharply, Mach objected that a container nobody can see is no explanation. Two centuries later relativity kept Newton's asymmetry and destroyed the other half of his scaffolding: absolute time.

Key takeaways

Newton's Principia (1687) posits two absolute background structures: absolute space, a rigid arena existing independently of matter, and absolute time, a universal duration flowing identically for all.
Absolute time means $t' = t$ in every inertial frame, which makes simultaneity absolute — a universal "now" spanning the whole universe.
Uniform velocity through absolute space is undetectable. Newton's own laws are boost-invariant ( $\mathbf{a}' = \mathbf{a}$ ), so no mechanical experiment can find absolute rest.
Acceleration is detectable. Spin water at $\omega$ and its surface becomes the paraboloid $z = \omega^2 r^2/2g$ : at $r = 0.10\ \mathrm{m}$ , $\omega = 10\ \mathrm{rad/s}$ the rim stands $5.1\ \mathrm{cm}$ above the centre. Absolute space announces itself only through acceleration.
Mach's reply: the water rotates relative to the fixed stars, not to an invisible container — inertia is a relation between matter and matter.
Key idea: absolute space is invisible to uniform velocity and visible to acceleration; that asymmetry is Newton's whole case. Absolute time, the other half of the scaffolding, is exactly what special relativity later destroys.

The core insight

Newtonian physics rests on two hidden postulates: space is a fixed arena, and time flows universally. Together they make the distance between simultaneous events and the duration between any two events frame-independent, with a global "now". But the arena is coy — uniform motion through it leaves no trace, while rotation leaves a fingerprint you can measure with a bucket of water. Newton read that asymmetry as proof the arena is real; Mach read it as proof that inertia is a relation to distant matter.

Where this lives today

Every aircraft, submarine and spacecraft carries a device that settles Newton's asymmetry to nine decimal places. A ring-laser gyroscope sends two counter-propagating beams around a closed optical loop; if the instrument rotates, the beams take different times to go round, and their beat frequency reports the rotation rate directly. No external reference is used or needed — the box is sealed, exactly as Newton's thought-experiment demanded, and the ring lasers at the Wettzell geodetic observatory read Earth's spin to parts in $10^{9}$ . An accelerometer beside it can measure acceleration, but no instrument in the box, of any design, can measure the vehicle's velocity. That is Newton's asymmetry shipped as hardware: an inertial navigation system knows how it has turned and how it has accelerated, but it must be told how fast it was going to begin with. Absolute time gets the same treatment one storey up. GPS receivers solve for position by comparing broadcast timestamps — a calculation wrong by kilometres per day if it assumes a single universal clock. Newton's second postulate is the one the system is built to correct.

The physical picture

Picture an infinite sheet of graph paper filling all space, motionless, there before any matter and remaining if all matter vanished: that is absolute space. Now imagine one master clock whose ticks define "now" everywhere at once: absolute time. Every position and every instant is read against these two unmoving references. Newton's argument is subtler than it looks. Run through the four stages and watch which quantity tracks the concavity. At the start bucket and water are both at rest — flat surface, no relative motion. Release the cord and the bucket spins while the water lags: the relative motion is now greatest, and the surface is still perfectly flat. Friction then brings the water up to speed, so the two turn together with no relative motion at all — and this is precisely when the surface is most hollow. Finally stop the bucket: relative motion returns, and the surface stays hollow. Relative rotation and concavity have no correlation whatever. What the water responds to is its own rotation, full stop.

Figure

Newton's bucket, four stages against the fixed grid of absolute space. The surface is flat when bucket and water spin relative to each other, and concave when they do not — so the concavity tracks the water's rotation in absolute space, never its rotation relative to the bucket.

Newton pressed the point with a second thought-experiment, the two globes: two spheres joined by a cord, alone in an otherwise empty universe, with no stars and nothing at all to be moving relative to. Set them spinning about their midpoint and the cord pulls taut, so measuring the tension measures the rotation — $T = m\omega^2 r$ , which for two $1\ \mathrm{kg}$ globes on a $2\ \mathrm{m}$ cord ( $r = 1\ \mathrm{m}$ ) turning at $3\ \mathrm{rad/s}$ reads a perfectly definite $\mathbf{9\ N}$ . The tension is quadratic in $\omega$ and vanishes only when $\omega = 0$ , so a spring balance alone distinguishes a rotating universe from a still one: in a universe with nothing to refer to, rotation still has a number. Mach's reply is that the question is empty — with no other matter there would be no inertia, hence no tension at all, and in the real universe the globes turn with respect to the distant stars. Now set the shy half beside it. Load the bucket onto a train cruising at any constant velocity: the surface stays exactly flat, and no experiment inside will tell you how fast you are going, because $\mathbf{F} = m\mathbf{a}$ is untouched by a boost. Absolute space, if it exists, punishes acceleration and ignores velocity — and the figure below puts a number on both halves.

Figure

The punchline: spin a bucket of radius r = 0.10 m at ω = 10 rad/s and the free surface becomes the paraboloid z = ω²r²/(2g), lifting the rim 5.1 cm above the vertex — the rim rises 2.5 cm above the still-water line while the centre sinks 2.5 cm below it. Boost the same bucket at any constant velocity and the surface stays perfectly flat.

The core equations

Absolute time is the shorter half to write down and the more consequential to lose: every inertial observer reads the same clock, and — crucially — the time coordinate carries no dependence on where the event happened. Absolute space then makes the map between frames a rigid shift, preserving the distance between simultaneous events.

t' = t, \qquad \mathbf{r}' = \mathbf{r} - \mathbf{v}\,t \qquad (1)

where: $t,\mathbf{r}$ — an event in frame $S$ ; $t',\mathbf{r}'$ — the same event in $S'$ , moving at constant $\mathbf{v}$ . Physical reading: because $t'=t$ contains no $\mathbf{r}$ , events simultaneous in $S$ are simultaneous in $S'$ — the universal "now". This is the line special relativity erases; its replacement lives under Relativity of Simultaneity. Differentiate twice and the shy half of absolute space appears: velocities differ between frames, $\mathbf{u}' = \mathbf{u} - \mathbf{v}$ , but the constant $\mathbf{v}$ dies under the second derivative.

\mathbf{a}' = \frac{\mathrm{d}^2\mathbf{r}'}{\mathrm{d}t'^2} = \frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2} = \mathbf{a} \qquad (2)

where: $\mathbf{a},\mathbf{a}'$ — acceleration of a particle in $S$ and $S'$ . Physical reading: Newtonian forces depend on separations and relative velocities, so $\mathbf{F}' = \mathbf{F}$ too, and $\mathbf{F} = m\mathbf{a}$ holds identically in both frames. No mechanical experiment can measure your velocity through absolute space — Newton said so himself, as Corollary V of the Principia. Now the loud half. In a frame co-rotating with the water, a surface element feels gravity pulling down at $g$ and a centrifugal effect pushing outward at $\omega^2 r$ . A liquid surface cannot support a shear, so it settles perpendicular to that combined effective gravity — fixing its slope at every radius.

\frac{\mathrm{d}z}{\mathrm{d}r} \;=\; \frac{\omega^2 r}{g} \qquad (3)

where: $z(r)$ — height of the surface above its lowest point; $r$ — distance from the axis; $\omega$ — angular velocity; $g = 9.81\ \mathrm{m\,s^{-2}}$ . Physical reading: the surface tilts more steeply the further out you go, in proportion to $\omega^2 r$ — the fingerprint of rotation, which no change of velocity can produce. Integrate that slope outward with $z(0)=0$ and the shape is forced. Not a cone, not a hemisphere, but a paraboloid — and the number it writes on the bucket wall is a direct reading of $\omega$ against absolute space.

\boxed{\;z(r) \;=\; \frac{\omega^2 r^2}{2g}\;} \qquad (4)

where: symbols as in Eq. (3). Physical reading: the rise is quadratic in both $r$ and $\omega$ , so doubling the spin quadruples the climb; for a bucket of radius $0.10\ \mathrm{m}$ at $10\ \mathrm{rad/s}$ the rim stands $5.1\ \mathrm{cm}$ above the vertex. Rotation is absolute in exactly this operational sense — the water reports it without consulting anything outside the bucket. The Earth is a slowly-spinning bucket, and Foucault's pendulum is the classic way to read its rotation from indoors: only the component of the planet's spin along the local vertical can turn the swing plane.

\Omega \;=\; \omega_\oplus \sin\varphi, \qquad T = \frac{2\pi}{\Omega} \qquad (5)

where: $\Omega$ — precession rate of the swing plane; $\omega_\oplus = 7.292\times10^{-5}\ \mathrm{rad\,s^{-1}}$ ; $\varphi$ — latitude; $T$ — time to turn full circle. Physical reading: at the pole the plane turns once per sidereal day ( $23.93\ \mathrm{h}$ ); at the equator it never turns; at $\varphi = 45^\circ$ it takes $33.9\ \mathrm{h}$ . The pendulum keeps its plane fixed with respect to something — and to a very good approximation that something is the fixed stars.

Where the asymmetry comes from

Derivation AThe rotating bucket's paraboloid, and why velocity leaves no trace

The goal is to derive the shape of the spinning water surface from first principles and then show, in the same breath, that the same physics forbids any comparable fingerprint of uniform motion. That double result is Newton's entire case for absolute space: it announces itself through acceleration and stays invisible to velocity. We work in the frame co-rotating with the water at angular velocity $\omega$ , where the fluid is at rest and its surface is in equilibrium.

Step 1. Account for the forces on a small element of the surface at radius $r$ : gravity, $-g$ downward, and the centrifugal effect $+\omega^2 r$ outward. Their sum is the effective gravity $\mathbf{g}_{\rm eff} = (\omega^2 r,\, -g)$ , which tilts further from the vertical the further out you go. Why: in the co-rotating frame the water is static, so the only way to describe its equilibrium is to fold the rotation into an effective gravitational field.

Step 2. Impose the condition defining a free surface: a liquid cannot support a shear stress, so at equilibrium the surface must lie perpendicular to $\mathbf{g}_{\rm eff}$ — otherwise a component of effective gravity would lie along it and the water would still be flowing. Perpendicularity means the tangent $(1,\ \mathrm{d}z/\mathrm{d}r)$ has zero dot product with $\mathbf{g}_{\rm eff}$ , giving $\mathrm{d}z/\mathrm{d}r = \omega^2 r/g$ . Why: this is the same rule that makes still water level in a stationary bucket, where $\mathbf{g}_{\rm eff}$ points straight down. Rotation simply tilts the field.

Step 3. Integrate outward from the axis with $z(0) = 0$ : $z(r) = \omega^2 r^2/2g$ . A paraboloid — fixed entirely by $\omega$ and $g$ , with no free constants and no reference to the bucket, the room or the Earth. With $r = 0.10\ \mathrm{m}$ , $\omega = 10\ \mathrm{rad/s}$ , $g = 9.81\ \mathrm{m\,s^{-2}}$ the rim sits $(100 \times 0.01)/(2\times 9.81) = 0.051\ \mathrm{m}$ above the vertex. Why: the water has just told you its rotation rate to two figures, with no window and no external landmark. That is what "absolute" means operationally.

Step 4. Now do the same for a boost. Put the apparatus on a train moving at constant $\mathbf{v}$ . By Eq. (2), $\mathbf{a}' = \mathbf{a}$ : the boost adds no term to the equation of motion, so $\mathbf{g}_{\rm eff}$ is unchanged and the slope picks up no $v$ -dependence. The surface stays exactly flat, for any $v$ , forever. Why: a boost is linear in $t$ and dies under the second derivative. Rotation is not — $\omega^2 r$ survives, and no choice of $\mathbf{v}$ can cancel it.

Step 5. Read the asymmetry. Rotation, an absolute acceleration, leaves a fingerprint detectable from entirely inside the closed system; uniform velocity leaves none and is undetectable in principle. Newton concluded the arena is real and that rotation is rotation with respect to it. Mach replied that the water rotates with respect to the fixed stars, and that in a genuinely empty universe there would be no fingerprint because there would be no inertia. Both agree on the equation and disagree about what $\omega$ is measured against. Why: this is the fault line that runs through the next two centuries and reappears in general relativity as frame-dragging. *These results check out: perpendicularity of the free surface to the effective gravity gives $\mathrm{d}z/\mathrm{d}r = \omega^2 r/g$ ; integrating with $z(0)=0$ gives $z = \omega^2r^2/2g$ ; the rim rise evaluates to $0.0510\ \mathrm{m}$ ; and $\mathbf{a}'=\mathbf{a}$ under the Galilean map.*

Go deeper · Derivation B · 1 minAbsolute time implies absolute simultaneity

Derivation BAbsolute time implies absolute simultaneity

The second pillar of the Scholium is easier to state and far harder to give up. Newton's absolute time gives every event a single number $t$ that all observers agree on, and Newtonian kinematics inherits that agreement wholesale. What we want to see precisely is why a universal clock forces a universal "now", and which feature of the transformation would have to change for simultaneity to become observer-dependent. That feature is the presence of $\mathbf{r}$ on the right-hand side of the time equation — and its absence is the assumption special relativity discards.

Step 1. Take events $A$ and $B$ at $(t_A,\mathbf{r}_A)$ and $(t_B,\mathbf{r}_B)$ in $S$ , and call them simultaneous there if $\Delta t = t_B - t_A = 0$ . Transform to a frame $S'$ moving at any constant $\mathbf{v}$ using Eq. (1): $t'_A = t_A$ and $t'_B = t_B$ , whatever the events' positions. Why: the Galilean time rule is $t' = t$ with no $\mathbf{r}$ anywhere in it — that is the entire content of "absolute" time.

Step 2. Subtract: $\Delta t' = t_B - t_A = \Delta t$ , so $\Delta t = 0$ if and only if $\Delta t' = 0$ . Simultaneity is frame-independent and "now" is a well-defined slice through the whole universe: you may ask whether a flare on Andromeda happened at the same instant as your heartbeat, and there is one answer. Why: $\Delta t'$ is independent of $\Delta \mathbf{r}$ , so the spatial separation plays no part — however far apart the events, and however fast $S'$ moves.

Step 3. Locate the hinge. Suppose the time rule carried a term proportional to spatial separation, $t' = t - (\text{something})\cdot\mathbf{r}$ . Then two events with $\Delta t = 0$ but $\Delta\mathbf{r} \neq 0$ would have $\Delta t' \neq 0$ : simultaneity would depend on the observer's motion, and the universal "now" would dissolve. Special relativity does exactly this. Why: the replacement is not derived here — see Relativity of Simultaneity and The Lorentz Transformation. The point is that absolute simultaneity is an assumption, not a logical necessity, and this is precisely where it enters. *These results check out: $t'=t$ gives $\Delta t' = \Delta t$ for all events, with no dependence on $\Delta\mathbf{r}$ , and $\vert\Delta\mathbf{r}'\vert = \vert\Delta\mathbf{r}\vert$ at fixed $t$ .*

Limiting cases

Case	Result	Meaning
$\omega \to 0$	$z \to 0$ , surface flat	The fingerprint vanishes precisely when the rotation does — nothing else can flatten it
Any boost $\mathbf{v}$	$\mathbf{a}' = \mathbf{a}$ , surface unchanged	Uniform velocity through absolute space is undetectable, by Newton's own laws
Slow spin ( $\omega^2 R \ll g$ )	rim rise tiny but non-zero	Earth's equatorial term $\omega_\oplus^2 R_\oplus = 0.034\ \mathrm{m\,s^{-2}}$ is $0.35\%$ of $g$ — enough to bulge the planet
Foucault, $\varphi=0$ vs $90^\circ$	$\Omega = 0$ ; $T = 23.93\ \mathrm{h}$	At the equator the swing plane never turns; at the pole it turns once per sidereal day — the cleanest reading of Earth's rotation
Everyday speeds ( $v \ll c$ )	$t' = t$ to fantastic accuracy	Absolute time is an outstanding approximation — which is why it felt like a fact for two centuries; its replacement is developed under Relativity of Simultaneity

Common trapThe rim does not rise 5.1 cm above the water line you started with

The formula $z = \omega^2 r^2/2g$ gives the height of the rim above the vertex — the lowest point of the spinning surface, at the centre. It does not say the water climbs $5.1\ \mathrm{cm}$ up the wall from where it sat before you started spinning. Water is conserved: whatever climbs the wall must come out of the middle. Averaging $z(r)$ over the disc gives $\omega^2R^2/4g$ , exactly half the rim rise, so the still-water line sits halfway up the parabola: the rim climbs $2.5\ \mathrm{cm}$ above the original level while the centre sinks $2.5\ \mathrm{cm}$ below it, and the two add back to $5.1\ \mathrm{cm}$ . Predict a $5.1\ \mathrm{cm}$ climb up the wall and you overshoot by a factor of two. The neighbouring trap: the surface is a paraboloid, not a cone and not a hemisphere, because the slope grows linearly with $r$ .

Worked examples

1 · The rotating bucket

A bucket of radius $r = 0.10\ \mathrm{m}$ is spun about its axis at $\omega = 10\ \mathrm{rad/s}$ until the water turns with it. How far does the rim of the surface stand above the centre, and where does the original water line sit?

Step 1. Apply the paraboloid, $z = \omega^2 r^2 / 2g$ , with $\omega = 10\ \mathrm{rad/s}$ , $r = 0.10\ \mathrm{m}$ , $g = 9.81\ \mathrm{m\,s^{-2}}$ : $z = (100 \times 0.01)/19.62 = 1/19.62$ .

Step 2. Evaluate: $z = 0.051\ \mathrm{m} = \boldsymbol{5.1\ \mathrm{cm}}$ . The rim stands $5.1\ \mathrm{cm}$ above the vertex at the centre — a hollow you can see across a room.

Step 3. Locate the original level. Averaging the paraboloid over the disc gives $\big(1/\pi R^2\big)\int_0^R (\omega^2r^2/2g)\,2\pi r\,\mathrm{d}r = \omega^2R^2/4g$ — exactly half the rim rise. So the still-water line sits $\boldsymbol{2.5\ \mathrm{cm}}$ below the rim and $2.5\ \mathrm{cm}$ above the vertex: the rim climbs $2.5\ \mathrm{cm}$ , the middle sinks $2.5\ \mathrm{cm}$ , and the two sum to $5.1\ \mathrm{cm}$ .

Step 4. Sense-check the steepness. At the rim the tilt is $\arctan(\omega^2 r/g) = \arctan(1.019) = 45.6^\circ$ — unmistakably visible. Halve the spin to $5\ \mathrm{rad/s}$ and the rise collapses to a quarter, $1.27\ \mathrm{cm}$ , because $z \propto \omega^2$ .

2 · The Foucault pendulum at latitude 45°

A long pendulum is hung at latitude $\varphi = 45^\circ$ and set swinging in a fixed vertical plane. How long does the plane of the swing take to turn full circle, and how fast does it rotate?

Step 1. Only the vertical component of Earth's angular velocity can turn the plane, so $\Omega = \omega_\oplus \sin\varphi$ . With $\omega_\oplus = 7.292\times10^{-5}\ \mathrm{rad\,s^{-1}}$ and $\sin 45^\circ = 0.7071$ , $\Omega = 5.156\times10^{-5}\ \mathrm{rad\,s^{-1}}$ .

Step 2. Convert to a period: $T = 2\pi/\Omega = 1.219\times10^{5}\ \mathrm{s}$ .

Step 3. In hours, $T = 1.219\times10^{5}/3600 = 33.85\ \mathrm{h} \approx \boldsymbol{33.9\ \mathrm{h}}$ — the familiar "a day divided by $\sin\varphi$ " ( $24\ \mathrm{h}/\sin 45^\circ = 33.9\ \mathrm{h}$ ; the sidereal $23.93\ \mathrm{h}$ gives $33.85\ \mathrm{h}$ , the same to three digits). The plane creeps round at $360/33.85 = 10.6^\circ$ per hour, easily visible in an afternoon.

Step 4. Check the extremes and Foucault's own case. At the pole the period is one sidereal day, $23.93\ \mathrm{h}$ ; at the equator $\sin\varphi = 0$ and the plane never turns. Foucault swung his in Paris, latitude $48.85^\circ$ , where theory gives $T = 31.8\ \mathrm{h}$ or $11.3^\circ$ per hour — and that is what the crowd in the Panthéon watched in 1851, indoors, with no view of the sky.

Physics of the portrait

The Scholium to the Definitions is no aside — it is the metaphysical foundation Newton thought his mechanics required, and he defended it with two physical experiments rather than with argument alone.

Origins · 1 minIsaac Newton and how this was discovered

Isaac Newton

1642–1727 · Woolsthorpe · Cambridge · London

Isaac Newton was born at Woolsthorpe in Lincolnshire and sent up to Trinity College, Cambridge, where a plague closure in 1665–66 sent him home for two years — the annus mirabilis in which he laid out the calculus, the composition of white light and the inverse-square law of gravitation, then told almost nobody for two decades. The Principia of 1687, written at Halley's prodding and expense, opens with the Definitions and their Scholium, where Newton distinguishes absolute from relative motion and offers the bucket and the two globes as evidence that rotation, at least, is real and not merely relational. He was scrupulous about the limits of his claim: Corollary V states outright that the motions of bodies among themselves are the same whether their common frame is at rest or moving uniformly — conceding that absolute velocity is undetectable. Lucasian Professor, then Master of the Mint (where he pursued counterfeiters with relish), and President of the Royal Society from 1703 until his death, he fought Leibniz over the calculus and wrote more on alchemy and biblical chronology than on physics.

History

1687

— Newton's Principia: the Scholium to the Definitions sets out absolute space and absolute time, with the bucket and the two globes as evidence for absolute rotation.

1715

— The Leibniz–Clarke correspondence begins; Leibniz argues that space is only the order of coexisting things, so an absolute container is superfluous.

1721

— George Berkeley's De Motu rejects absolute space, time and motion as unobservable fictions: all motion is motion relative to other bodies.

1851

— Léon Foucault demonstrates his pendulum, first at the Paris Observatory in February and then publicly in the Panthéon in March — Earth's rotation read from indoors.

1883

— Ernst Mach's Die Mechanik in ihrer Entwicklung attacks the bucket argument and proposes that inertia arises from the distant matter of the universe.

1905

— Einstein's relativity of simultaneity removes absolute time.

1908

— Minkowski's Cologne lecture Raum und Zeit fuses space and time into one geometry.

Going further

Texts. Newton, Philosophiæ Naturalis Principia Mathematica (1687), Scholium to the Definitions (Cohen and Whitman translation, 1999) · Mach, The Science of Mechanics, Ch. 2 §6 (the critique of the bucket) · Sklar, Space, Time, and Spacetime, Ch. 3 · Taylor and Wheeler, Spacetime Physics, 2nd ed., Ch. 1. Key numerical results: perpendicularity of the free surface to the effective gravity gives $\mathrm{d}z/\mathrm{d}r = \omega^2 r/g$ , and integrating gives $z = \omega^2r^2/2g$ ; the rim rise is $0.051\ \mathrm{m}$ at $r=0.10\ \mathrm{m}$ , $\omega=10\ \mathrm{rad/s}$ , $g=9.81\ \mathrm{m\,s^{-2}}$ , with the still-water level splitting it into $\pm2.5\ \mathrm{cm}$ and a rim tilt of $45.6^\circ$ ; the two-globe tension is $9\ \mathrm{N}$ ; $\mathbf{a}'=\mathbf{a}$ and $\Delta t'=\Delta t$ under the Galilean map; and $T = 2\pi/(\omega_\oplus \sin 45^\circ) = 1.219\times10^{5}\ \mathrm{s} = 33.9\ \mathrm{h}$ , with $23.93\ \mathrm{h}$ at the pole and $31.8\ \mathrm{h}$ at Paris ( $48.85^\circ$ ). Anchor values used throughout: $g = 9.81\ \mathrm{m\,s^{-2}}$ , $\omega_\oplus = 7.292\times10^{-5}\ \mathrm{rad\,s^{-1}}$ , $R_\oplus = 6.378\times10^{6}\ \mathrm{m}$ .

Builds on. Inertial Frames of Reference · The Principle of Relativity. Leads to. Galilean Relativity · Galilean Transformations · The Luminiferous Ether · The Constancy of the Speed of Light.

If uniform motion through absolute space cannot be detected, why did Newton keep absolute space at all?

Because he thought acceleration forced it on him. Newton conceded the velocity point explicitly — Corollary V says the motions of bodies among themselves are the same whether their common frame is at rest or moving uniformly. But rotation is different: the bucket hollows and the cord between the globes pulls taut, whether or not anything nearby is available to be rotating relative to. Something must be rotating with respect to something, and Newton could see no candidate but space itself. The move is honest, and its weakness is what Mach seized on — it explains an observable effect by an unobservable cause.

Doesn't the bucket just show that the water is rotating relative to the Earth?

That is the natural modern reflex, and it is not what the experiment says. The concavity does not depend on the Earth being there — it depends only on $\omega$ , and appears identically in a bucket spun on a spacecraft. What the bucket demonstrates is the absence of any correlation between concavity and rotation relative to the nearest available body, the bucket wall: the surface is flattest at stage II, when that relative rotation is greatest. Mach's counter is not "relative to the Earth" but "relative to the fixed stars" — the distant matter of the universe — a much larger and much less easily dismissed claim.

What exactly did Mach claim, and was he right?

Mach's claim, in Die Mechanik (1883), was that inertia is not a property a body has by itself but a relation to all the other matter in the universe: the water climbs the wall because it rotates relative to the fixed stars, and in a universe emptied of stars there would be no centrifugal effect and no concavity. It inspired Einstein, who coined the phrase "Mach's principle". Whether general relativity honours it is still debated: the theory does show that rotating matter drags local inertial frames with it — frame-dragging, measured by Gravity Probe B and by laser-ranging satellites — which is Machian in flavour, but it also admits rotating solutions in otherwise empty universes, which is not.

Why is the spinning water surface a parabola rather than a cone?

Because the sideways pull grows with radius. A cone needs a constant slope, hence a constant sideways force everywhere; but the centrifugal effect is $\omega^2 r$ , zero on the axis and growing linearly outward. The surface stays perpendicular to the total effective gravity, so its slope is $\mathrm{d}z/\mathrm{d}r = \omega^2 r/g$ — proportional to $r$ , not constant — and integrating a linearly growing slope gives a quadratically growing height, $z = \omega^2r^2/2g$ . This is also why liquid-mirror telescopes work: spin a dish of mercury and nature grinds you a perfect parabolic mirror for free.

Isn't absolute simultaneity just obviously true?

It feels obvious for a specific reason: every speed in your life is minute compared with the speed of light, and the effects that break it scale as $v/c^2$ times a separation. At walking pace, across a room, the disagreement about "now" is far below anything your clocks could register, so the universal present is an outstanding approximation and your intuition was trained on it. But an excellent approximation is not a logical necessity: as the derivation above shows, absolute simultaneity follows from one feature of the Galilean map — that $t' = t$ carries no dependence on position — and a feature is something experiment is free to overturn. It did.

Do we still use absolute space and time for anything?

Constantly, and with a clear conscience. Every bridge, every orbit, every engine and every weather model is computed in a Newtonian framework with a single universal time, because at the speeds involved the corrections sit far below the noise. What changed is the framework's status: an excellent approximation with a known regime of validity, not the true stage of the world. The dividing line is set by precision — atomic clocks and satellite navigation are the everyday technologies that first crossed it, and they must correct for effects a universal clock cannot describe.