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Uniform low-speed expansion of the observer energy, momentum and kinetic energy of a massive four-momentum

Statement

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives)), $c > 0$ , let $n \in V$ be future unit with $η (n, n) = - 1$ , and let $P = m U$ with $m > 0$ and $U \in V$ future timelike, $η (U, U) = - c^{2}$ . Let $E_{n} (P)$ , $q_{n} (P)$ , $T_{n} (P)$ and $m (P)$ be the observer energy, spatial momentum, kinetic energy and invariant mass of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum, put $h (v) : = v + η (n, v) n$ as in Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c), and let $γ : = - η (U / c, n) \geq 1$ and $w : = h (U) / γ = c h (U / c) / γ \in n^{⊥}$ be the Lorentz factor and relative velocity of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(d); then $u : = U / c$ satisfies $u = γ (n + w / c)$ , $∣ w ∣^{2} = c^{2} (1 - γ^{- 2})$ and $γ = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ , and Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum,(b) gives $E_{n} (P) = m c^{2} γ$ , $q_{n} (P) = m γ w$ , $T_{n} (P) = m c^{2} (γ - 1)$ and $m (P) = m$ .

Let $z_{0} \in [0, 1)$ , suppose $z : = ∣ w ∣^{2} / c^{2} \leq z_{0}$ , put $K (z_{0}) : = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ , and let $f, R$ be as in The uniform second-order Taylor remainder of $z \mapsto (1 - z)^{- 1 / 2}$ on $[0, z_{0}]$ , so that $γ = f (z)$ and $γ = 1 + \frac{z}{2} + R (z)$ with $∣ R (z) ∣ \leq K (z_{0}) z^{2}$ . Then:

(a) $E_{n} (P) = m c^{2} + \frac{1}{2} m ∣ w ∣^{2} + R_{E}$ and $T_{n} (P) = \frac{1}{2} m ∣ w ∣^{2} + R_{E}$ for the common real number $R_{E} : = m c^{2} R (z)$ , and $∣ R_{E} ∣ \leq K (z_{0}) z m ∣ w ∣^{2} = K (z_{0}) \frac{m ∣ w ∣^{4}}{c^{2}}, ∣ T_{n} (P) - \frac{1}{2} m ∣ w ∣^{2} ∣ \leq 2 K (z_{0}) z \frac{1}{2} m ∣ w ∣^{2} .$

(b) $q_{n} (P) = m w + R_{q}$ with $R_{q} : = m (γ - 1) w$ , and $∣ R_{q} ∣ \leq m ∣ w ∣ (\frac{z}{2} + K (z_{0}) z^{2}) \leq (\frac{1}{2} + K (z_{0}) z_{0}) z m ∣ w ∣;$ moreover $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1} = m γ ∣ w ∣$ and $m (P) = m$ .

(c) Every constant above depends only on $z_{0}$ , so the estimates are uniform over all future unit $n$ , all $m > 0$ and all $P = m U$ with $∣ w ∣^{2} \leq z_{0} c^{2}$ ; if $z = 0$ (equivalently $w = 0$ , $U = c n$ ) then $R_{E} = R_{q} = 0$ , $E_{n} (P) = m c^{2}$ , $q_{n} (P) = 0$ and $T_{n} (P) = 0$ . The lemma is a statement about the mathematical model only; it asserts no measured quantity, no approximation of a physical theory and no error estimate for any experiment.

Facts & Assumptions

Given: The affine Minkowski model $(A, V, η, C^{+})$ , $c > 0$ , a future unit $n$ , $P = m U$ with $m > 0$ and $U$ future timelike with $η (U, U) = - c^{2}$ , the projection $h (v) = v + η (n, v) n$ , the numbers $γ = - η (U / c, n)$ and $w = h (U) / γ$ , a bound $z_{0} \in [0, 1)$ with $z : = ∣ w ∣^{2} / c^{2} \leq z_{0}$ , and $K (z_{0}) = \frac{3}{8} (1 - z_{0})^{- 5 / 2}$ .

[L1]

The observer quantities: $E_{n} (P) = - c η (P, n)$ , $q_{n} (P) = h (P) \in n^{⊥}$ , $m (P) = \sqrt{- η (P, P)} / c$ for nonzero future causal $P$ , and $T_{n} (P) = E_{n} (P) - m c^{2}$ after the constant $m > 0$ is fixed; $P = (E_{n} (P) / c) n + q_{n} (P)$ is the unique decomposition of $P$ along $n$ and $n^{⊥}$ (Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum).

[L2]

Projection and relative velocity: $h$ is linear with image $n^{⊥}$ , kernel $s p a n (n)$ , $h (n) = 0$ , and the restriction of $η$ to $n^{⊥}$ is positive definite with norm $∣ w ∣ = \sqrt{η (w, w)}$ ; for future unit $u$ one has $γ = - η (n, u) \geq 1$ , $w = c h (u) / γ \in n^{⊥}$ , $u = γ (n + w / c)$ , $∣ w ∣^{2} = c^{2} (1 - γ^{- 2})$ , $γ = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ , and $γ = 1$ if and only if $u = n$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c),(d)).

[L3]

Observer identities: for $P = m U$ with $m > 0$ constant and $U$ future timelike, $η (U, U) = - c^{2}$ , and $γ, w$ as above, one has $E_{n} (P) = m c^{2} γ$ , $T_{n} (P) = m c^{2} (γ - 1) \geq 0$ , $q_{n} (P) = m γ w$ , $m (P) = m$ and $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1} = m γ ∣ w ∣$ (Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum,(b)).

[L4]

Uniform Taylor remainder: for $z_{0} \in [0, 1)$ and $0 \leq z \leq z_{0}$ , $f (z) = (1 - z)^{- 1 / 2} = 1 + \frac{z}{2} + R (z)$ with $∣ R (z) ∣ \leq K (z_{0}) z^{2}$ ; in particular $f (0) = 1$ and $R (0) = 0$ (The uniform second-order Taylor remainder of $z \mapsto (1 - z)^{- 1 / 2}$ on $[0, z_{0}]$ ).

[L5]

Model and powers: the model data are those of Oriented time-oriented affine Minkowski space (the spacetime primitives); $∣ w ∣^{2} = η (w, w) \geq 0$ on $n^{⊥}$ and $z = ∣ w ∣^{2} / c^{2} \geq 0$ ; rational powers such as $(1 - z)^{- 1 / 2}$ are those of Rational powers $a^{r}$ of a positive base and integer powers those of Integer powers $a^{m}$ .

[L6]

Scalar absolute values satisfy $∣ a b ∣ = ∣ a ∣ ∣ b ∣$ , $∣ a ∣ \geq 0$ , and $∣ a ∣ = a$ for $a \geq 0$ (Absolute value in an ordered field, Basic properties of the absolute value). The rest-space norm supplied by [L2] (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c)) satisfies $∣ λ v ∣ = ∣ λ ∣ ∣ v ∣$ for $v \in n^{⊥}$ .

Proof

technique · direct

1.1givenL2L4L5algebra

Relative velocity and the Taylor form of $γ$ . The map $h$ is linear by [L2], so $h (U) = c h (U / c)$ and $w = h (U) / γ = c h (U / c) / γ$ ; the vector $u : = U / c$ is future unit and $γ = - η (n, u) \geq 1$ , so [L2],(d) gives $w \in n^{⊥}$ , $u = γ (n + w / c)$ , $∣ w ∣^{2} = c^{2} (1 - γ^{- 2})$ and $γ = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2} = (1 - z)^{- 1 / 2} = f (z)$ . By [L4] applied to $z \leq z_{0}$ we have $γ = 1 + \frac{z}{2} + R (z)$ with $∣ R (z) ∣ \leq K (z_{0}) z^{2}$ ; define $R_{E} : = m c^{2} R (z)$ .

1.2givenL3algebra

The observer identities. By [L3] at these data, $E_{n} (P) = m c^{2} γ$ , $T_{n} (P) = m c^{2} (γ - 1)$ , $q_{n} (P) = m γ w$ , $m (P) = m$ and $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1} = m γ ∣ w ∣$ ; define $R_{q} : = q_{n} (P) - m w = m (γ - 1) w$ .

1.3givenL5L6algebra

Translating $z$ into norms. Since $z = ∣ w ∣^{2} / c^{2}$ , we have $m c^{2} z = m ∣ w ∣^{2}$ , and also $K (z_{0}) z^{2} m c^{2} = K (z_{0}) z m ∣ w ∣^{2} = K (z_{0}) m ∣ w ∣^{4} / c^{2}$ . Both expressions are nonnegative because $m > 0$ , $K (z_{0}) > 0$ and $∣ w ∣^{2} \geq 0$ .

2.1step 1.1step 1.2step 1.3L1algebra

Part (a), the expansions. By steps 1.1 and 1.2, $E_{n} (P) = m c^{2} (1 + \frac{z}{2} + R (z)) = m c^{2} + \frac{1}{2} m ∣ w ∣^{2} + R_{E}$ using $m c^{2} \frac{z}{2} = \frac{1}{2} m ∣ w ∣^{2}$ from step 1.3, and $T_{n} (P) = E_{n} (P) - m c^{2} = \frac{1}{2} m ∣ w ∣^{2} + R_{E}$ by the definition of $T_{n}$ in [L1].

2.2step 1.1step 1.3L4L6algebra

Part (a), the bounds. By step 1.1, $∣ R_{E} ∣ = m c^{2} ∣ R (z) ∣ \leq m c^{2} K (z_{0}) z^{2} = K (z_{0}) z m ∣ w ∣^{2} = K (z_{0}) m ∣ w ∣^{4} / c^{2}$ using step 1.3; the second displayed bound is $∣ T_{n} (P) - \frac{1}{2} m ∣ w ∣^{2} ∣ = ∣ R_{E} ∣ \leq K (z_{0}) z m ∣ w ∣^{2} = 2 K (z_{0}) z \cdot \frac{1}{2} m ∣ w ∣^{2}$ .

2.3step 1.1step 1.2L4L6algebra

Part (b). By step 1.2, $q_{n} (P) = m w + R_{q}$ with $R_{q} = m (γ - 1) w$ . Norm homogeneity [L6] gives $∣ R_{q} ∣ = m ∣ γ - 1 ∣ ∣ w ∣$ ; since $γ = f (z)$ , the bound on $∣ f (z) - 1 ∣$ of The uniform second-order Taylor remainder of $z \mapsto (1 - z)^{- 1 / 2}$ on $[0, z_{0}]$ ,(b) gives $∣ R_{q} ∣ \leq m ∣ w ∣ (\frac{z}{2} + K (z_{0}) z^{2}) \leq m ∣ w ∣ z (\frac{1}{2} + K (z_{0}) z_{0}),$ where the final inequality uses $z \leq z_{0}$ and $K (z_{0}) > 0$ . The norm identity $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1} = m γ ∣ w ∣$ and $m (P) = m$ are those of step 1.2.

3.1step 2.1step 2.2step 2.3L1L2L4algebra

Uniformity and the case $z = 0$ . The remainders $R_{E} = m c^{2} R (z)$ and $R_{q} = m (γ - 1) w$ and the bounds of steps 2.2 and 2.3 depend on the state only through $z$ , $m$ and $∣ w ∣$ , and their constants depend on $z_{0}$ alone, so the estimates hold uniformly over all future unit $n$ , all $m > 0$ and all $P = m U$ with $∣ w ∣^{2} \leq z_{0} c^{2}$ . If $z = 0$ then $γ = (1 - z)^{- 1 / 2} = 1$ by step 1.1, so $u = n$ and $w = 0$ by [L2],(d), whence $R (0) = 0$ by [L4], $R_{E} = 0$ , $R_{q} = 0$ , and steps 1.2 and 2.1 give $E_{n} (P) = m c^{2}$ , $q_{n} (P) = 0$ , $T_{n} (P) = 0$ ; conversely $w = 0$ gives $z = 0$ , and $U = c n$ is the same case.

4.1step 3.1given∎

Conclusion. Steps 2.1 and 2.2 are clauses (a), step 2.3 is clause (b) including the norm identity and $m (P) = m$ , and step 3.1 is clause (c) with the uniformity statement and the $z = 0$ case; no step used more than the stated hypotheses $z \leq z_{0} < 1$ , $m > 0$ and the model's normalization, and the statement concerns the mathematical model only.

Uniform low-speed expansion of the observer energy, momentum and kinetic energy of a massive four-momentum

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources