← All parts of this equation

Equation 104 · Part 8 · The Clock That Comes Back Wrong by Exactly Its Mass

Denominator: hbar

Δϕclock=−ΔEℏ Δτ.\Delta\phi_{\rm clock}=-\frac{\Delta E}{\hbar}\,\Delta\tau.
ℏ\hbar

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

hbar occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

The proper-time loop LP\mathcal L_{\rm P} is where mass stops being a fixed label and becomes something an object can carry a little more or less of depending on what it is doing internally. A composite system with an internal excitation of energy Δ\Delta E above its ground configuration carries additional rest mass Δ\Delta E/c2c^2 , by ordinary mass–energy equivalence — no redefinition of either the relativistic or the everyday meaning of mass, simply the standard bookkeeping in which a system’s total rest energy Mc2c^2 includes whatever internal energy HintH_{\rm int} it holds: M=M0M_0+HintH_{\rm int}/c2c^2 . If the same object is sent along two paths that differ in proper time by Δ\Deltaτ\tau , and its…

Read this part in the article →

Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.