REL-29

Four-momentum norm

On-shell energy.

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GoverningFour-momentum norm

Governing equation

E^2-p^2 c^2=m^2 c^4

$p \in \left[1, 5000\right]$

where

p: p (MeV/c)
m: m (MeV/c²)
E: Four-momentum norm (MeV)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

E^2-p^2 c^2=m^2 c^4

$p \in \left[1, 5000\right]$

$p = 1$ $E = 1.123$

Along the curve

②

Domain of the variable — blue

p_{\min} = 1,\quad p_{\max} = 5000

$p \in \left[1, 5000\right]$

$p = 1$ $p = 1$

Along the curve

③

Response curve — green

E^2-p^2 c^2=m^2 c^4

$p \in \left[1, 5000\right]$

$p = 1$ $E = 1.123$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E}{\mathrm{d}p}

$p \in \left[1, 5000\right]$

$p = 32.44$ $E^{\prime} = 0.998$

Along the curve

Lecture brief

Historical brief

Einstein’s 1905 Lorentz kinematics and 1915 field equation, then Schwarzschild (1916) and Hawking temperature, recast time, mass and gravity. The lab computes dilation, E=mc² and horizon scales. This sheet (REL-29 — Four-momentum norm) is the form associated with Four-momentum norm. Working symbols: $p$ , $m$ $\rightarrow$ $E$ . On-shell energy. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

E

from

p

m

in Relativity via

E^2-p^2 c^2=m^2 c^4

On-shell energy. Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

p = 200.000\,\mathrm{MeV/c}

m = 0.511\,\mathrm{MeV/c^{2}}

, the governing relation

E^2-p^2 c^2=m^2 c^4

yields

E = 200.001\,\mathrm{MeV}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

p p200 MeV/cm m0.511 MeV/c²

Outputs

Four-momentum norm E200.001 MeV

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

On-shell energy. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube