General Relativity in 50 Minutes: The One Equation Extension

 

Instructor: J. Rogers

Prerequisites: The 50‑minute SR class (energy‑momentum invariant, $\gamma$, $E=\gamma m_0{c}^2$, etc.)
Goal: Show that the core prediction of GR (gravitational time dilation) follows directly from SR + Newtonian escape velocity.

Opening Hook (3 minutes)

“Last time we learned that special relativity is one equation: $E^2=(pc{)}^2+(m_0{c}^2{)}^2$. Today I’ll show you that general relativity’s most famous effect — gravitational time dilation — is just that same equation with one new idea: escape velocity. ”

Write on board:

$$\text{GR}=\text{SR}+\text{escape velocity}$$

Step 1: Escape velocity from Newton (5 minutes)

Recall from classical physics: to escape a planet of mass $M$ from radius $r$, you need kinetic energy ≥ potential energy:

$$\frac{1}{2}{m}_0{v}_{\text{esc}}^2=\frac{GM{m}_0}{r}$$

Solve:

$$v_{\text{esc}}^2=\frac{2GM}{r}$$

Define ${\beta }_{\text{esc}}=v_{\text{esc}}\mathrm{/}c$:

$${\beta }_{\text{esc}}^2=\frac{2GM}{r{c}^2}$$

“This is a pure number, independent of the test mass $m_0$. It’s the gravitational potential in dimensionless form.”

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Step 2: Plug escape velocity into the Lorentz factor (10 minutes)

Recall the Lorentz factor from SR:

$$\gamma =\frac{1}{\sqrt{1-{\beta }^2}}$$

Now, instead of $\beta$ being the actual velocity of a moving object, interpret $\beta$ as the escape velocity at that radius.

If you are at rest at radius $r$ (i.e., not falling), the escape velocity is the speed needed to reach infinity. GR tells us that time runs slower in a gravitational well. Plug ${\beta }_{\text{esc}}^2$ into $\gamma$:

$${\gamma }_{\text{grav}}=\frac{1}{\sqrt{1-\frac{2GM}{r{c}^2}}}$$

But careful: in GR, the time dilation factor is $1\mathrm{/}{\gamma }_{\text{grav}}$, not ${\gamma }_{\text{grav}}$. Because a clock at rest at radius $r$ ticks slower than a clock at infinity. So:

$$\frac{d\tau }{dt}=\sqrt{1-\frac{2GM}{r{c}^2}}=\frac{1}{{\gamma }_{\text{grav}}}$$

“That’s the famous Schwarzschild time dilation. It’s just the reciprocal of the SR Lorentz factor, with the escape velocity playing the role of relative speed.”

Step 3: Derive gravitational redshift (5 minutes)

From time dilation, a photon emitted at radius $r$ with frequency $f_{\text{em}}$ will be observed at infinity with lower frequency:

$$f_{\mathrm{\infty }}=f_{\text{em}}\sqrt{1-\frac{2GM}{r{c}^2}}\approx f_{\text{em}}(1-\frac{GM}{r{c}^2})$$

“This is a direct prediction, verified by the Pound–Rebka experiment. No Einstein equations needed — just SR + escape velocity.”

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Step 4: The “other half” — spatial curvature (10 minutes)

“But GR also predicts light bending and Mercury’s perihelion shift. Those come from spatial curvature, not just time dilation. However, we can get the order of magnitude using the same escape velocity.”

For light grazing the Sun, the deflection angle (in radians) is approximately $2GM\mathrm{/}(R_{\odot }{c}^2)$ — the same as ${\beta }_{\text{esc}}^2$ at the Sun’s surface. The full GR prediction is twice that (due to spatial curvature). So we can write:

$$\text{Deflection}=(1+\underset{\text{from spatial curvature}}{\underbrace{1}})\cdot {\beta }_{\text{esc}}^2$$

“Thus, the extra factor of 2 tells us that space is curved as well. But the core scale is still ${\beta }_{\text{esc}}^2$, the same number that appeared in time dilation.”

Step 5: The unified picture (7 minutes)

Write on board:

SR (relative motion)	GR (gravity)
$\gamma =1\mathrm{/}\sqrt{1-{\beta }^2}$	$1\mathrm{/}\gamma =\sqrt{1-{\beta }_{\text{esc}}^2}$
$\beta =v\mathrm{/}c$	${\beta }_{\text{esc}}^2=2GM\mathrm{/}(r{c}^2)$

“Notice: the math is identical — only the interpretation of $\beta$ changes. In SR, $\beta$ is the relative speed between two observers. In GR, ${\beta }_{\text{esc}}$ is the escape velocity from a gravitational potential. That’s why GR is a trivial extension of SR: you just replace the kinetic β with the gravitational β.”

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Step 6: Why we don’t need full Einstein equations (5 minutes)

“The full Einstein field equations are complex because they describe how mass–energy curves spacetime in every direction. But for spherically symmetric, static situations (like planets, stars, black holes), the Schwarzschild solution — which contains exactly this $\sqrt{1-2GM\mathrm{/}(r{c}^2)}$) factor — is enough. And we just derived that factor from SR + escape velocity.”

“Thus, for most practical purposes (GPS, gravitational time dilation, light deflection, black hole thermodynamics), you don’t need the full machinery. You need only this one extension.”

Closing (5 minutes)

“So here’s the takeaway: General relativity is not a separate, mysterious theory. It’s special relativity applied to the escape velocity of a gravitational field. These are just two sides of one coin. The same Lorentz factor gives time dilation in both cases; gravity just swaps $\gamma$ for $1\mathrm{/}\gamma$ because the clock slows down instead of speeding up. And the natural scale is set by ${\beta }_{\text{esc}}^2=2GM\mathrm{/}(r{c}^2)$.”

“Next time someone says GR is impossibly hard, show them this. One new number — escape velocity — plugged into SR’s Lorentz factor gives the core prediction of GR.”

Assessment Questions

1. Derive the gravitational time dilation factor using escape velocity and the Lorentz factor.

2. For a neutron star with $M=1.4M_{\odot }$ and $R=10$ km, compute ${\beta }_{\text{esc}}^2$ and the time dilation factor at its surface.

3. Explain why the deflection of light by the Sun is twice the Newtonian prediction.

Why This Works Pedagogically

• Conceptual continuity: Students already know $\gamma$ from SR. They just reinterpret $\beta$.

• No tensor calculus: The core physical effect (time dilation) is derived with algebra.

• Reveals unity: GR is not a “paradigm shift” but a natural extension — exactly as Einstein himself saw when he derived gravitational time dilation from the equivalence principle and SR.