Foundations — General relativity — equivalence principle, curved spacetime (overview)

Before you can read the parent note on general relativity, you must be able to look at every symbol in it and see a picture. This page takes them one at a time, from the most ordinary (mass, acceleration) to the frightening ($g_{\mu \nu }$, $G_{\mu \nu }$). Nothing here assumes you have met any of it before. If you can add, multiply, and imagine a ball on a floor, you can follow every line.

This is the foundations page for the parent overview.

---

1. The two kinds of mass — the seed of everything

Everyone knows one word "mass". Physics secretly uses it for two different jobs, and the whole of general relativity begins when we notice those two jobs give the same number.

The rule that defines it is Newton's second law:

Here two new symbols appeared, so let us earn them:

• $F$ = force = a push or pull, measured in newtons. Picture an arrow: longer arrow = harder push.
• $a$ = acceleration = how fast the speed itself is changing (speeding up, slowing down, or turning). Picture a car's speedometer needle sweeping — the speed of that sweep is $a$.

Now the second job of mass:

The gravity rule that defines it:

$F=m_{g}g$

where the lowercase $g$ = the strength of the gravitational field (on Earth's surface, about $9.8{\text{m/s}}^2$). Picture $g$ as "how many metres-per-second of downward speed gravity hands you every second."

Set the two definitions equal for a freely falling object (the only force is gravity, $F=m_{g}g$, and Newton says $F=m_{i}a$):

$m_{i}a=m_{g}g⟹a=\frac{m_g}{m_i}g.$

Because $m_g/m_i=1$ for everything, $a=g$ for everything — the mass cancels. A feather and a hammer fall together. That cancellation is the door into GR. The figure below shows both objects, drawn with different sizes, tracing identical falls:

Look at the two downward arrows: they reach the floor at the same instant even though the amber "hammer" is far more massive than the cyan "feather." The little box in the middle records why — the ratio $m_g/m_i$ is $1$, so the mass never enters the answer.

---

2. Reading the "change" symbols: $\Delta$, ratios, and squares

The parent note is full of expressions like $\frac{1}{2}g{t}^2$. Let us disarm each piece before we ever meet a formula that uses them.

• $t$ = time, seconds. A stopwatch.
• $t^2$ = $t$ times $t$. Why does time get squared in falling? Because in free fall the distance grows faster and faster — after twice the time you have fallen four times as far. That "four for two" pattern is exactly what squaring does.

Where the $\frac{1}{2}$ in $\Delta y=\frac{1}{2}g{t}^2$ comes from

This one-half is not a fudge factor — you can see it. An object dropped from rest has speed $0$ at the start and speed $gt$ at time $t$ (because each second gravity adds $g$ to the speed). Plot speed against time and you get a straight line climbing from $0$ up to $gt$.

Distance fallen = area under that speed line (speed $\times$ time, added up). But the graph is a triangle, not a rectangle: base $t$, height $gt$. A triangle's area is $\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}t(gt)=\frac{1}{2}g{t}^2$. That triangle is exactly where the one-half is born — it is the "half" in "area of a triangle."

Compare the shaded triangle with the dashed rectangle in the figure: if the object had moved at its final speed the whole time it would cover the rectangle's area; because it started slow, it covers only half — the triangle.

Here a new symbol $h$ appears in the parent formulas. Let us pin it down:

---

3. Frequency $f$ — why light carries a "clock"

The figure shows a wave emitted low down (tight crests, high $f$) and the same wave received higher up (stretched crests, lower $f$) after climbing against gravity:

Follow the white arrow upward: the cyan wave at the top has fewer crests per stretch than the amber wave at the bottom — that visible stretching is the redshift, and equally, the "clock" ticking slower.

Now that $f$ exists, the symbol $\Delta f$ from Section 2 gets its meaning: $\Delta f$ = the change in frequency between emission and reception. The fraction $\Delta f/f$ (change compared to the whole) is what the parent's redshift formula predicts — but we cannot write that formula until one more symbol, $c$, is on the table.

---

4. Proper time $\tau$ — the clock a traveller actually reads

So now both deferred symbols are earned: $\Delta f$ = change in frequency, $\Delta \tau$ = change in a clock's reading. The redshift statement will tie them together: a stretched-out wave ($\Delta f<0$) and a slow clock ($\Delta \tau$ running behind) are the same fact.

---

5. The speed of light $c$, and why it is squared everywhere

Why does $c^2$ (a gigantic number) sit in the denominator of expressions like $gh/c^2$? Because relativity's effects are measured relative to how close things move to $c$. Dividing by the enormous $c^2$ is why gravitational time shifts are usually tiny — but "tiny" is not "zero," which is why GPS still needs them.

---

6. Building the redshift formula $\Delta f/f=-gh/c^2$ from the pieces

Now every symbol in the parent's key formula is defined: $g$ (field strength, §1), $h$ (height, §2), $f$ (frequency, §3), $c$ (light-speed, §5), $\Delta$ and the ratio idea (§2). Let us build the formula so it is a story, not a spell.

Step 1 — give the photon an effective mass (WHAT & WHY). A photon has energy $E=hf$. By $E=m{c}^2$, that energy behaves like a mass $m=E/c^2=hf/c^2$. We do this because gravity acts on anything with mass-energy, so we need a "gravitational handle" on the photon.

Step 2 — make it climb and pay the gravitational toll (WHAT & WHY). Lifting a mass $m$ through height $h$ in field $g$ costs energy $mgh$ (this is the everyday "lift-it-higher costs work" rule). The photon has no pocket to pay from except its own energy, so it loses that much: $\Delta E=-mgh=-\frac{hf}{c^2}gh.$ The minus sign says "energy went down on the way up."

Step 3 — turn lost energy into lost frequency (WHAT & WHY). Since $E=hf$, a change in energy is a change in frequency: $\Delta E=h\Delta f$. We use this because $f$ is what we actually observe as colour/redshift. Substitute: $h\Delta f=-\frac{hf}{c^2}gh.$

Step 4 — cancel and read off the fraction (WHAT IT LOOKS LIKE). Planck's $h$ cancels on both sides; divide by $f$:

Why exactly this size? Two ingredients set it: the toll per unit energy is $gh/c^2$ (gravity's pull $g$ times the climb $h$, measured against light's speed-squared), and frequency is proportional to energy, so the fractional energy loss and the fractional frequency loss are one and the same number. The figure of §3 is this formula's picture: crests stretch by exactly this fraction.

---

7. Reference frames, free-fall, and "weightless"

---

8. Tidal effects, geodesics, and curvature — the geometry words

Inside one small box you can erase gravity by free-falling. But drop two balls far apart above the Earth: both aim at Earth's centre, so their paths slowly lean toward each other.

The figure shows two free-fallers starting side by side high above Earth; watch their amber paths lean inward toward the planet's centre:

The dotted white lines point straight at Earth's centre — since both fallers aim there, they must converge. No re-choice of "still room" can remove that leaning-together; that stubborn convergence is curvature made visible.

---

9. The metric $g_{\mu \nu }$ and the field equations $G_{\mu \nu }=T_{\mu \nu }$

These are the scariest symbols in the parent note. Tamed, they are just bookkeeping tables.

First: what do the indices $\mu$ and $\nu$ mean?

Second: the shorthand that hides a sum

Third: the metric itself

The line element written with it, $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }(\text{sum over }\mu ,\nu =0,1,2,3),$ is just an accountant's total unpacked by the summation convention: $ds$ is the true little distance, the $dx$'s are the little coordinate steps, and each $g_{\mu \nu }$ weights the matching pair before all sixteen are added. In flat spacetime it collapses to the familiar Pythagoras-with-a-time-term you meet in Spacetime Metric & Minkowski Diagram.

Fourth: the field equations

You are not expected to compute these on this page — only to see them as "curvature table = matter table," so the parent note's boxed equation is a sentence, not a wall. The machinery lives in Geodesics & Curvature.

---

Prerequisite map

The diagram below traces how each foundation feeds the next, ending at the parent overview. Read it top-to-bottom: the two masses meet in the equivalence principle; light-and-frequency plus that principle give time dilation and bending; free-fall plus tidal effects give curvature; curvature gives the metric and field equations — and everything flows into GR.

---

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, reread that section before opening the parent note.