Gravity, free fall, and clocks

1.1 Why does a falling scale read zero?#

Stand on a spring scale. Its spring compresses because it supports you. The scale measures the force with which it pushes on your feet; its familiar kilogram display converts that force using Earth’s usual surface gravity. Your mass and this supporting force are different quantities.

Now imagine you and the scale inside a cabin that is falling freely. This is a thought experiment: ignore air resistance, rotation, and any contact with the outside. You, the scale, and the cabin all fall together. The scale no longer needs to support you, so its reading drops to zero. Your mass has not vanished. Neither has Earth.

This is weightlessness: the absence of a supporting force. Free fall means motion with no rocket thrust, floor, or other nongravitational push. It need not mean falling straight down; an orbiting spacecraft can be in free fall too.

An accelerometer makes a related measurement. Imagine a small test mass suspended by springs inside a case. When the case pushes the mass away from its natural free fall, the springs deform. Their deformation gives an acceleration reading. On the ground the reading is about $9.8\,\mathrm{m/s^2}$; in ideal free fall it is zero. We call this instrument reading proper acceleration. Acceleration obtained by differentiating a position on a ground-based map answers a different question. A falling ball has nonzero downward acceleration on that map while its ideal accelerometer reads zero.

So far, a small falling cabin can imitate a cabin drifting far from significant gravity. There is a more revealing experiment. Release two small test balls at rest relative to one another, with no springs joining them. Ignore their mutual attraction. Track the distance between them over time.

If the balls start side by side at the same height near Earth, their downward directions point toward the same center: they tend to approach one another. If one starts directly above the other, the lower one falls more strongly, so their separation tends to grow. These are tidal effects: differences in free-fall acceleration across a region. The effect is very small in a small cabin over a short time, but it is a measurable prediction.

Predict before continuing: could a zero scale reading establish that gravity is absent?

Compare your reasoning

No. The scale tests whether it must support you. Comparing two freely falling test balls tests whether free fall changes from place to place. You can have zero support and still detect a changing relative acceleration. One observer can carry out this comparison; the experiment requires two test bodies, not two people with different powers of observation.

General relativity will describe these tidal effects using spacetime curvature. For now, that name points to an experiment we can state without knowing its mathematics. Chapters 8–10 build and measure the corresponding geometry. The small-region qualification is essential to the equivalence principle: a finite falling room does not remove tidal effects. Einstein Online: the equivalence principle.

1.2 Comparing clocks#

Ordinary mechanics treats rulers and clocks as a fixed background: we first agree on distances and times, then use forces to predict motion. Relativity asks us to examine the measuring procedures too.

Imagine two identical clocks meeting, separating, and later meeting again. They can compare their elapsed times directly at reunion. Could their readings depend on their journeys? Special relativity says yes, even when gravity is negligible. Chapter 3 will derive that prediction from the measured speed of light and a careful definition of distant clock synchronization.

Gravity adds another question. Hold one clock higher above Earth than the other and compare their rates using light signals. General relativity predicts a difference even though the clocks are stationary relative to the ground. Chapter 16 will calculate it, with the clocks and comparison procedure specified.

A theory of gravity must therefore predict both falling motion and clock comparisons. Spacetime is our name for considering where and when things happen together. An event is one occurrence with a place and a time, such as a particular flash or two clocks meeting. These words help us describe the experiments; the mathematical measuring rule comes later, in Chapters 3 and 4.

The point is already concrete: a path on a map does not tell the whole story. We must also ask what a clock carried along that path records.

1.3 What Einstein’s equation must predict#

Earth’s mass affects the motion of a nearby ball. The ball also contributes, by a much smaller amount, to the gravitational situation. A complete theory must describe this mutual influence.

Einstein’s equation is the rule connecting the distribution and motion of matter with the geometry that determines clock readings and free-fall behavior. Its source includes energy and pressure as well as mass. We have begun defining those quantities in Chapter 0; Chapter 11 will show how to collect the relevant measurements.

There is an essential limit to the slogan “matter determines geometry.” An empty region can still have tidal effects because matter lies elsewhere. Gravitational disturbances can also travel through an empty region. Specifying the local matter alone does not settle the whole problem: we need suitable starting and boundary information, just as an acceleration law alone did not specify a particular throw.

Chapter 12 will state the equation precisely, after we have built its ingredients. For now, its job is to connect the behavior of matter with a consistent account of clocks, light, and falling bodies.

1.4 Predicting motion and predicting gravity#

There are two different calculations ahead of us:

1. Given a gravitational situation, predict how a test clock, a light pulse, or a freely falling object behaves.
2. Determine the gravitational situation itself from matter and suitable starting or boundary information.

A test object is small enough that we can neglect its effect on the situation we are studying. Predicting the fall of one small ball near Earth usually uses this approximation. Predicting two stars orbiting each other requires accounting for both stars.

We must also describe how matter behaves. Does a gas resist compression? Does it exchange heat? The gravitational equation alone does not answer those questions. We supply physical models for the matter and solve them together with gravity. A successful calculation states those assumptions and ends with a measurement we could compare with an experiment.

1.5 What a rubber sheet can show#

You may have seen a heavy ball make a dip in a rubber sheet while smaller balls roll around it. The picture helps suggest that geometry can differ from a flat plane. It is a poor explanation of why gravity works.

The small balls roll because ordinary gravity pulls them downward; the demonstration already uses the effect it is meant to explain. It also leaves out clocks. A picture of space at one instant cannot show how elapsed times depend on a journey.

Nor does the sheet’s outside room belong to the theory. In Chapter 4 we will learn how inhabitants can study a surface’s geometry using measurements within it, without looking from outside.

Keep the falling cabin, the two test balls, and the two clocks as our starting experiments. A useful picture should help us predict one of those measurements. A distorted grid by itself is not evidence of gravity.

1.6 From free fall to a theory#

A falling cabin suggests that some effects we attribute to gravity can disappear when we change how the laboratory moves. The two-ball experiment shows the limit: neighboring falls can still converge or separate. A theory has to account for both observations at once.

It also has to recover the successful predictions of Newton’s theory where we have already tested them. Free fall is a clue toward a new theory, but it does not uniquely determine its equations. Clock comparisons, light propagation, and planetary motion provide further tests. We will return to these tests after developing the mathematics.

1.7 What comes next#

Our next task is to describe the same motion using different axes without changing what is being predicted. Chapter 2 introduces vectors and the measuring rules called covectors. Chapter 3 then uses light signals to compare clocks in relative motion.

These are the first steps toward a common account of clocks and free fall. The learning path shows where the later calculations fit.

1.8 Try the falling-cabin experiment#

Imagine three cabins. One rests on Earth, one falls freely near Earth, and one is far from significant gravity with a rocket accelerating it upward. The rocket gives its cabin the same accelerometer reading as the cabin on Earth. Each cabin contains a person standing on a scale.

Which scales read zero? What can the readings tell us?

The freely falling cabin’s scale reads zero. The ground and the rocket each push their cabin into the person’s feet, so those scales give a nonzero reading. With the stated accelerations, they give the same reading for the same person.

That scale reading alone cannot tell the person whether a planet is nearby. To investigate further, compare the motion of separated test objects or exchange signals with the outside. Near Earth, the two-ball experiment can reveal tidal effects even in the falling cabin.

1.9 How the theory developed#

The experiments above organize our route through the subject. The historical route involved years of revisions and collaboration.

Read the illustrated history: Newton to Einstein and beyond

The order in this book is designed for learning. Discovery followed a much less direct route. Keep these two stories separate: a clean derivation tells us how ideas fit together now; a historical account asks what the people involved actually knew then.

1687: one law for terrestrial and celestial motion. Newton’s Principia brought the motion of falling bodies and planets into the same mathematical account. His gravity was enormously successful. Its instantaneous interaction and the unexplained proportionality of inertial and gravitational mass later became productive questions, not reasons to dismiss its achievements.

1905–1908: clocks and geometry become inseparable. Einstein’s special relativity replaced universal simultaneity with a consistent account of measurements by moving observers. Minkowski organized the theory geometrically in four-dimensional spacetime. That did not yet make the geometry dynamical.

1907: free fall becomes the clue. Einstein recognized the special status of a freely falling observer. The equivalence principle suggested a link between acceleration, gravitational clock shifts, and gravity. It did not by itself supply the final field equation.

1912–1913: mathematical collaboration and a wrong turn. Marcel Grossmann helped Einstein bring differential geometry into the problem. Their Entwurf theory used a mathematical rule for distances and times but had restricted gravitational equations. Recovering Newtonian gravity and deciding which changes of description those equations should permit were entangled difficulties. The route was not “notice curvature, write the answer.”

November 1915: revision in public. Einstein presented successive communications on November 4, 11, 18, and 25. The November 18 calculation explained Mercury’s anomalous perihelion advance. The November 25 paper gave the final field equations. Hilbert was developing an action-based approach in the same period. The surviving documents matter more than a simple race narrative; a paper’s submission date and the content of its later printed version are different evidence. Einstein’s November 25 paper, Norton’s historical analysis.

After the field equation: interpretation remained hard. Schwarzschild’s spherical solution, expanding cosmological models, rotating black holes, gravitational radiation, and singularity theorems exposed consequences that were not obvious from the equation. Observations eventually made these subjects experimentally accessible. Chapter 18 follows one concrete landmark: the first direct gravitational-wave detection in 2015, reported in 2016. Chapter 19 explains what an expanding solution means before asking what data favor it.

Einstein and Grossmann’s work, including the unsuccessful 1913 theory, is examined in Janssen and Renn, Untying the Knot.

1.10 Watching neighboring objects fall#

The animation below draws lines between imagined small laboratories falling toward Earth. The laboratories move; the lines let us compare their positions. A changing grid does not mean that Earth consumes space.

Compare neighbors. Along the same outward radial line, the inner laboratory falls faster and the radial gap grows. Side-by-side laboratories fall toward the same center and their sideways gap shrinks. The grid makes the tidal pattern from §1.1 visible. Its motion is accelerated for display and stops at Earth’s surface.

The illustration chooses one particular family of falls. Each laboratory starts from rest infinitely far away, so it already has an inward speed when it enters the scene. It does not depict objects released from rest at the edge of the picture. Starting conditions matter here just as they did for the ball in Chapter 0.

In the Newtonian approximation, the speed follows from the energy calculation in §0.5. Starting from rest infinitely far away gives total energy zero. At distance $r$ from a spherical body of mass $M$,

Here $m$ is a test body’s mass and $v$ is its speed. For inward motion, distance from the center decreases, so $dr/dt=-v$. The test mass cancels: within this approximation, all of these laboratories obey the same speed rule.

The full relativistic model used by the animation has additional coordinate assumptions. Its numerical motion is not a general formula for every possible observer’s speed. We return to the distinction between chosen position labels and measured speeds in Chapters 16–17. The river-model paper by Hamilton and Lisle documents that model; it is further reading, not preparation for the next chapter.

What to read from the picture: compare how neighboring falls change relative to one another. The connecting lines are a drawing aid. Their bend alone does not establish curvature. The ScienceClic visualization by Alessandro Roussel motivates the moving-grid presentation.

---

SPACETIME LAB / 01

Compare two falling objects

Follow the rose points as they fall toward Earth. Watch the gap between points on the same radial line, then compare points side by side.

Interactive geometry is loading.

$$\frac{dr}{dt}=-\sqrt{\frac{2G_NM}{r}}$$

Read the scene. The teal grid joins freely falling observers; rose points make their motion visible. This model treats Earth as a nonrotating sphere. Each observer has fallen from rest very far away and is already moving when it enters the picture. New points enter from beyond the view, and the motion stops at the surface. The grid lines simply join observers at the same time.

How this animation is calculated

Each rose point follows the speed rule derived in §1.10. Integrating that rule tells us where to draw the point at a later time:

$$\begin{gathered}\frac{dr}{dt}=-\sqrt{\frac{2G_NM}{r}}\\r(t)=\left[r_0^{3/2}-\frac32\sqrt{2G_NM}\,t\right]^{2/3}\end{gathered}$$

$r_0$ is the distance from Earth’s center when the point enters, and $t$ is the time since entry. $M$ is Earth’s mass; $G_N$ is Newton’s gravitational constant. The formula applies until the point reaches the surface.

The points show a continuous stream of falls with these starting conditions. The connecting lines help you compare their positions. They are neither material threads nor the paths of individual objects.

Motion is shown at 110 times real time. The model uses Earth’s mean radius and gravitational parameter, giving an incoming surface speed of about $11.2\,\mathrm{km/s}$. It omits rotation, the atmosphere’s resistance, and Earth’s interior. The glow and lighting identify the globe; they carry no gravity data.

This speed rule also occurs in a useful relativistic description of spherical gravity. Chapter 17 develops the geometry needed to understand that description. The technical source is Hamilton and Lisle’s river model.

Presentation inspired by ScienceClic’s visualization. Earth texture: NASA, Blue Marble: Next Generation. Sources and image rights ↗

WORKED EXAMPLE

Release two pebbles, not just one

What can a falling laboratory detect without looking outside?

Builds onInitial data Mechanics

See the idea

An ideal accelerometer attached to a freely falling test particle reads zero. That does not say what neighboring freely falling particles do. A pair of pebbles can drift apart even while both accelerometers read zero. Their changing separation is the experiment that survives moving into a falling frame.

Work it out

1. 01
   Compare the two accelerations

   In a Newtonian one-dimensional approximation let $z$ measure upward position and let acceleration vary as $g(z)=-g_0+\kappa z$. Here $g_0$ is a constant acceleration and $\kappa$ is its change per unit height, in $\mathrm{s^{-2}}$. If two freely falling particles have positions z and $z+\xi$, subtract their equations of motion.

   $$\ddot\xi=g(z+\xi)-g(z)=\kappa\xi.$$

   Why this step works The shared acceleration cancels; its spatial variation remains.

2. 02
   Change the common acceleration

   Move to an accelerating coordinate system $z'=z-b(t)$. Each coordinate acceleration loses the same $\ddot b$. The difference is unaffected.

   $$\ddot z'_2-\ddot z'_1=\ddot z_2-\ddot z_1.$$

   Why this step works Subtracting the same number from both readings cannot change their difference.

3. 03
   Turn a small effect into a measurement

   For initially stationary separation $\xi_0$, a short-time expansion gives $\xi(t)=\xi_0+\kappa\xi_0t^2/2+\cdots$. The dots start at fourth order in time for this constant-$\kappa$ model with zero initial relative velocity. The leading formula needs $|\kappa|t^2\ll1$. A real comparison also requires distances small enough that the assumed linear field is accurate.

   $$\frac{\Delta\xi}{\xi_0}\simeq\frac12\kappa t^2.$$

   Why this step works The observation has a size and time scale; “local” is a controlled limit.

Go deeper

This is a Newtonian calibration, not the GR derivation. Chapter 10 will derive the relativistic description of this separation experiment. A rocket pushing a cabin through otherwise empty space can produce an accelerometer reading too. One such reading does not identify its cause. Comparing separated test masses asks whether their free-fall accelerations differ; following a single mass does not supply that comparison.

Further readingDavid Tong · General relativity

Test the idea

FIRST, PREDICT

Both pebble-mounted accelerometers read zero. Must their separation remain constant?

No. Neighboring free-fall trajectories can have relative acceleration.Yes. Zero accelerometer readings eliminate all gravitational effects.No, because one of the accelerometers must secretly be wrong.

Compare the reasoning

No. Neighboring free-fall trajectories can have relative acceleration.

Weightlessness of each pebble does not eliminate spatial differences in the gravitational field.

Yes. Zero accelerometer readings eliminate all gravitational effects.

One local reading cannot measure the relative motion of neighboring trajectories.

No, because one of the accelerometers must secretly be wrong.

Both local readings and the changing separation can be correct.

A hint

Separate the reading at one event from a comparison between nearby histories.

NOW CHANGE THE EXAMPLE

Take $\kappa=0.02\,\mathrm{s^{-2}}$, $\xi_0=3\,\mathrm m$ and initially zero relative velocity. What is the leading separation change after $1\,\mathrm s$?

A hint

Use $\Delta\xi\simeq\kappa\xi_0t^2/2$.

Work through the solution

$\Delta\xi\simeq0.02(3)/2=0.03\,\mathrm m$.

Weightlessness is a local accelerometer reading; tides compare neighboring free-fall histories.