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Counterexample: no speed-independent uniform low-speed remainder for the kinetic energy

Statement refuted

False claim. Let $c > 0$ and $m > 0$ . There is a real $B \geq 0$ such that $∣ T_{n} (P) - \frac{1}{2} m ∣ w ∣^{2} ∣ \leq B$ for every future unit $n \in V$ , every $P = m U$ with $U$ future timelike and $η (U, U) = - c^{2}$ , and the relative velocity $w$ of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(d), where $T_{n}$ is the kinetic energy of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum.

Counterexample. Fix a future unit $n$ and a unit vector $e \in n^{⊥}$ once; for every integer $N \geq 2$ use that same vector (possible because $n^{⊥}$ is a three-dimensional real subspace on which the restriction of $η$ is positive definite, so every nonzero vector of $n^{⊥}$ may be divided by its norm, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c)) and put $w_{N} : = c \sqrt{1 - N^{- 2}} e, U_{N} : = c N (n + \frac{w_{N}}{c}), P_{N} : = m U_{N} .$ Then $∣ w_{N} ∣^{2} = c^{2} (1 - N^{- 2}) < c^{2}$ , and $η (U_{N}, U_{N}) = c^{2} N^{2} (- 1 + ∣ w_{N} ∣^{2} / c^{2}) = - c^{2}$ with $U_{N}$ future timelike, so $U_{N}$ has the required four-velocity normalization; $u_{N} = U_{N} / c$ is future unit timelike; the relative velocity and Lorentz factor of $U_{N}$ are $w_{N}$ and $N$ , and Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum,(b) gives $T_{n} (P_{N}) = m c^{2} (N - 1)$ together with $\frac{1}{2} m ∣ w_{N} ∣^{2} = \frac{1}{2} m c^{2} (1 - N^{- 2})$ . Hence $T_{n} (P_{N}) - \frac{1}{2} m ∣ w_{N} ∣^{2} = m c^{2} (N - \frac{3}{2} + \frac{1}{2 N^{2}}),$ which exceeds any prescribed real $B$ for all sufficiently large $N$ ; equivalently the relative deviation $\frac{T_{n} (P_{N}) - \frac{1}{2} m ∣ w_{N} ∣^{2}}{\frac{1}{2} m ∣ w_{N} ∣^{2}} = \frac{(2 N + 1) (N - 1)}{N + 1}$ is unbounded in $N$ . Therefore the refuted statement is false for every pair $(c, m)$ , and any valid low-speed error bound must depend on the admissible speed domain (the constants $K (z_{0})$ of Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities do). The construction is a statement about the stipulated model and contains no empirical claim.

Facts & Assumptions

Given: The affine Minkowski model $(A, V, η, C^{+})$ of Oriented time-oriented affine Minkowski space (the spacetime primitives), a real $c > 0$ and a real $m > 0$ , a future unit vector $n$ (a nonzero vector $v$ of $C^{+}$ divided by the positive square root $(- η (v, v))^{1 / 2}$ of the positive number $- η (v, v)$ , Rational powers $a^{r}$ of a positive base), and a unit vector $e \in n^{⊥}$ .

[L1]

The model is an oriented time-oriented affine Minkowski space, so $η$ has inertia $(3, 1, 0)$ , a vector is timelike when $η (v, v) < 0$ and spacelike when $η (v, v) > 0$ , and the time orientation $C^{+}$ is one of the two connected components of the timelike set, hence nonempty and closed under positive scalar multiples; a future unit vector $n$ has $n \in C^{+}$ and $η (n, n) = - 1$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones).

[L2]

Rest space and relative velocity: $n^{⊥} = {v : η (n, v) = 0}$ is a three-dimensional linear subspace on which $η$ restricts to a positive definite form with norm $∣ w ∣ = \sqrt{η (w, w)}$ ; for a future unit $u$ , $γ = - η (n, u) \geq 1$ , $w = c h (u) / γ \in n^{⊥}$ with $h (u) = u + η (n, u) n$ , $u = γ (n + w / c)$ , $∣ w ∣^{2} = c^{2} (1 - γ^{- 2})$ and $γ = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ (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, $η (U, U) = - c^{2}$ , one has $E_{n} (P) = m c^{2} γ$ , $T_{n} (P) = m c^{2} (γ - 1)$ and $q_{n} (P) = m γ w$ , where $γ, w$ are the relative Lorentz factor and velocity of $U$ and $T_{n} (P) = E_{n} (P) - m c^{2}$ is the kinetic energy after the parameter $m$ is fixed (Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum,(b), Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum).

[L4]

Powers and absolute values: for $x > 0$ the rational power $x^{1 / 2}$ is the positive square root, and negative integer powers are the special case of Rational powers $a^{r}$ of a positive base on positive bases; $∣ x ∣ \geq 0$ , $∣ x y ∣ = ∣ x ∣ ∣ y ∣$ and $∣ x ∣ = x$ for $x \geq 0$ (Absolute value in an ordered field, Basic properties of the absolute value).

[L5]

Archimedean property: for every real $x$ there is a natural number $n \geq 1$ with $x < n \cdot 1_{R}$ , so an integer $N > x$ may be chosen (Every complete ordered field is Archimedean).

[L6]

Every path in a topological space has both endpoints in one connected component (Every path-connected space is connected, and every path component lies inside a component, clause 3).

Counterexample

technique · direct

1.1givenL1L2L4

The fixed data and the family. Fix once and for all the future unit $n$ and the unit vector $e \in n^{⊥}$ ; such data exist because timelike vectors exist by the inertia $(3, 1, 0)$ of the form and any nonzero timelike vector can be normalized by dividing by the positive rational power $(- η (v, v))^{1 / 2}$ , and because $n^{⊥}$ is a three-dimensional positive definite space by [L2]. For every integer $N \geq 2$ set $w_{N} : = c \sqrt{1 - N^{- 2}} e$ , $U_{N} : = c N (n + w_{N} / c)$ and $P_{N} : = m U_{N}$ , using the same $n$ and $e$ for every $N$ .

2.1step 1.1L1L2L6algebra

The normalization of $U_{N}$ . Since $e \in n^{⊥}$ is unit, $η (e, e) = ∣ e ∣^{2} = 1$ and $η (n, w_{N}) = c \sqrt{1 - N^{- 2}} η (n, e) = 0$ , so $η (w_{N}, w_{N}) = c^{2} (1 - N^{- 2}) = ∣ w_{N} ∣^{2} < c^{2}$ . Hence $η (U_{N}, U_{N}) = c^{2} N^{2} (η (n, n) + 2 η (n, w_{N} / c) + η (w_{N} / c, w_{N} / c)) = c^{2} N^{2} (- 1 + (1 - N^{- 2})) = - c^{2},$ and the continuous segment $a (s) = n + s w_{N} / c$ , $0 \leq s \leq 1$ , has $η (a (s), a (s)) = - 1 + s^{2} (1 - N^{- 2}) < 0$ throughout. The segment is continuous in any transported Euclidean norm because $∣ a (s) - a (t) ∣_{E} = ∣ s - t ∣ ∣ w_{N} / c ∣_{E}$ . By [L6], its endpoint lies in the same component as $n \in C^{+}$ (Time orientation, future and past timelike/null/causal vectors, and the closed cones). Thus $n + w_{N} / c \in C^{+}$ and $U_{N} = c N (n + w_{N} / c)$ is future timelike by positive scaling.

3.1step 2.1L2algebra

The relative velocity and Lorentz factor of $U_{N}$ . The vector $u_{N} : = U_{N} / c = N (n + w_{N} / c)$ satisfies $η (u_{N}, u_{N}) = - 1$ by step 2.1, so $u_{N}$ is future unit and $γ_{N} : = - η (n, u_{N}) = - N η (n, n) - N η (n, w_{N}) / c = N \geq 1$ . By [L2], the relative velocity of $U_{N}$ is $w = c h (u_{N}) / γ_{N} = c N h (n + w_{N} / c) / N = c h (w_{N} / c) = w_{N}$ and its Lorentz factor is $N$ ; consistently $∣ w_{N} ∣^{2} = c^{2} (1 - N^{- 2}) = c^{2} (1 - γ_{N}^{- 2})$ .

4.1step 3.1L3L4algebra

The kinetic energy and the comparison value. By [L3] applied to $P_{N} = m U_{N}$ with Lorentz factor $γ_{N} = N$ from step 3.1, $T_{n} (P_{N}) = m c^{2} (N - 1), \frac{1}{2} m ∣ w_{N} ∣^{2} = \frac{1}{2} m c^{2} (1 - N^{- 2}),$ so $T_{n} (P_{N}) - \frac{1}{2} m ∣ w_{N} ∣^{2} = m c^{2} (N - 1 - \frac{1}{2} + \frac{1}{2 N^{2}}) = m c^{2} (N - \frac{3}{2} + \frac{1}{2 N^{2}}) .$ For $N \geq 2$ this difference is positive, hence equal to its absolute value: $N - \frac{3}{2} + \frac{1}{2 N^{2}} > N - \frac{3}{2} \geq \frac{1}{2} > 0$ for every $N \geq 2$ .

5.1step 4.1L4algebra

The relative deviation. The ratio of the difference of step 4.1 to the comparison value $\frac{1}{2} m ∣ w_{N} ∣^{2} = \frac{1}{2} m c^{2} (1 - N^{- 2})$ equals $\frac{N - \frac{3}{2} + \frac{1}{2 N^{2}}}{\frac{1}{2} (1 - N^{- 2})} = \frac{2 N^{3} - 3 N^{2} + 1}{N^{2} - 1} = \frac{(N - 1)^{2} (2 N + 1)}{(N - 1) (N + 1)} = \frac{(N - 1) (2 N + 1)}{N + 1} \geq N - 1,$ using $(N - 1)^{2} (2 N + 1) = 2 N^{3} - 3 N^{2} + 1$ and $(2 N + 1) \geq (N + 1)$ for $N \geq 0$ .

6.1step 4.1step 5.1L4L5algebra

Both sequences are unbounded. Let $B \geq 0$ be any real. By [L5] choose an integer $N > \max {2, B / (m c^{2}) + \frac{3}{2}}$ . Then by step 4.1, $∣ T_{n} (P_{N}) - \frac{1}{2} m ∣ w_{N} ∣^{2} ∣ = m c^{2} (N - \frac{3}{2} + \frac{1}{2 N^{2}}) > m c^{2} (N - \frac{3}{2}) > B,$ so the bound $B$ fails at the admissible state $(n, P_{N})$ ; and by step 5.1 the relative deviation at that $N$ is at least $N - 1 > B / (m c^{2}) + \frac{1}{2}$ , so it is unbounded as well. Since $B \geq 0$ was arbitrary and the construction uses one fixed future unit $n$ , the refuted claim fails for every pair $(c, m > 0)$ .

7.1step 6.1L1∎

Conclusion. The displayed family of states $P_{N} = m U_{N}$ in the stipulated model has $T_{n} (P_{N}) - \frac{1}{2} m ∣ w_{N} ∣^{2} \to \infty$ with the relative deviation also tending to infinity, so no real constant $B$ can bound $∣ T_{n} (P) - \frac{1}{2} m ∣ w ∣^{2} ∣$ uniformly over all admissible states; a valid low-speed error bound necessarily depends on the admissible speed domain, as the constants $K (z_{0})$ of Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities do, which is confirmed independently by the bounded $z \leq z_{0}$ hypothesis of The uniform second-order Taylor remainder of $z \mapsto (1 - z)^{- 1 / 2}$ on $[0, z_{0}]$ . No experimental, measurement or other empirical claim is made.

Counterexample: no speed-independent uniform low-speed remainder for the kinetic energy

Statement refuted

Facts & Assumptions

Counterexample

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