← Mathematical compendium

Published equation contexts

dτd\tau

Why this formula appears here

N⊥N_\perp[Γ\Gamma] is a real-valued, non-negative functional of a timelike worldline in a fixed background spacetime together with a stated local energy convention; because the integrand is an energy and dτ\tau is a time, the integral has action units and N⊥N_\perp is dimensionless. It is invariant under reparametrizing Γ\Gamma ’s own curve parameter, since only proper time enters, and invariant under any passive coordinate change of the ambient spacetime, since both dτ\tau and a comoving-frame energy are scalars. It is not invariant under boosting the local tetrad used to evaluate ⟨\langle H(τ\tau)⟩\rangle — a different local observer at the same event assigns a different local energy — which is why…

Read the full article-specific guide →

Read the representative guide

dd

Symbol d

d is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Read this term in its guide →

How to interpret it

Read this expression with the definitions, units, and assumptions supplied by the article.

Research cited beside this formula

Learn the mathematics behind it

Published contexts (2)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

dτd\tau

Equation 24 · Evolutionary Physics

A Horizon Is a Toll Booth, Not a Loophole

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

N⊥N_\perp[Γ\Gamma] is a real-valued, non-negative functional of a timelike worldline in a fixed background spacetime together with a stated local energy convention; because the integrand is an energy and dτ\tau is a time, the integral has action units and N⊥N_\perp is dimensionless. It is invariant under reparametrizing Γ\Gamma ’s own curve parameter, since only proper time enters, and invariant under any passive coordinate change of the ambient spacetime, since both dτ\tau and a comoving-frame energy are scalars. It is not invariant under boosting the local tetrad used to evaluate ⟨\langle H(τ\tau)⟩\rangle — a different local observer at the same event assigns a different local energy — which is why…

Meanings in this article

  • τ\tau: a time, the integral has action units.
Equation guide → · Article →
dτd\tau

Equation 27 · Evolutionary Physics

A Horizon Is a Toll Booth, Not a Loophole

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

N⊥N_\perp[Γ\Gamma] is a real-valued, non-negative functional of a timelike worldline in a fixed background spacetime together with a stated local energy convention; because the integrand is an energy and dτ\tau is a time, the integral has action units and N⊥N_\perp is dimensionless. It is invariant under reparametrizing Γ\Gamma ’s own curve parameter, since only proper time enters, and invariant under any passive coordinate change of the ambient spacetime, since both dτ\tau and a comoving-frame energy are scalars. It is not invariant under boosting the local tetrad used to evaluate ⟨\langle H(τ\tau)⟩\rangle — a different local observer at the same event assigns a different local energy — which is why…

Meanings in this article

  • τ\tau: a time, the integral has action units.
Equation guide → · Article →