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Positivity, mass shell and kinetic energy of the observer quantities of a 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 timelike, $η (n, n) = - 1$ , and let $E_{n}, q_{n}, m, T_{n}$ be the quantities of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum.

(a) If $P$ is nonzero future causal, then $E_{n} (P) > 0$ , $q_{n} (P) \in n^{⊥}$ and

$η (q_{n} (P), q_{n} (P)) = \frac{E_{n} (P)^{2}}{c^{2}} + η (P, P) \geq 0.$

Consequently: (i) if $P$ is future null, then $η (P, P) = 0$ and $E_{n} (P) = c ∣ q_{n} (P) ∣ > 0$ ; (ii) if $P$ is future timelike with invariant mass $m (P)$ , then $η (P, P) = - m (P)^{2} c^{2}$ and the mass-shell identity

$E_{n} (P)^{2} = m (P)^{2} c^{4} + c^{2} ∣ q_{n} (P) ∣^{2}$

holds. The invariant mass $m (P)$ depends only on the scalar $η (P, P)$ and therefore on no observer $n$ and no basis, while $E_{n}$ and $q_{n}$ do depend on $n$ .

(b) Let $P = m U$ with $m > 0$ constant, let $U$ be future timelike with $η (U, U) = - c^{2}$ (the four-velocity normalisation of Four-velocity and four-acceleration of a future timelike worldline), and put $γ : = - η (U / c, n) \geq 1$ and $w : = c h (U / c) / γ \in n^{⊥}$ , the relative velocity of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(d) applied to the future unit vector $u = U / c$ ; equivalently $w = h (U) / γ$ . Then

$E_{n} (P) = m c^{2} γ, T_{n} (P) = m c^{2} (γ - 1) \geq 0, q_{n} (P) = m γ w,$

and $m (P) = m$ . Moreover $T_{n} (P) = 0$ if and only if $γ = 1$ , that is $U = c n$ , equivalently $P = m c n$ (the particle is at rest relative to $n$ ). Finally $∣ q_{n} (P) ∣ = m γ ∣ w ∣ = m c \sqrt{γ^{2} - 1}$ .

Facts & Assumptions

Given: the affine Minkowski model $(A, V, η, C^{+})$ ; a real number $c > 0$ ; a future unit timelike $n \in V$ with $η (n, n) = - 1$ ; clause (a): a nonzero $P \in c l (C^{+})$ ; clause (b): a real $m > 0$ and a future timelike $U \in V$ with $η (U, U) = - c^{2}$ , with $P = m U$ .

[F1]

The definition of the observer quantities (Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum, Definition): for $P \in c l (C^{+})$ one has $E_{n} (P) = - c η (P, n)$ and $q_{n} (P) = h (P) = P + η (P, n) n \in n^{⊥}$ , and $η (n, P) = - E_{n} (P) / c$ ; for nonzero future causal $P$ the invariant mass is $m (P) = \sqrt{- η (P, P)} / c$ ; for $P = m U$ with $m > 0$ and $η (U, U) = - c^{2}$ the kinetic energy is $T_{n} (P) = E_{n} (P) - m c^{2}$ ; and $∣ w ∣ = \sqrt{η (w, w)}$ on $n^{⊥}$ .

[F2]

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

[F3]

Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space: $η$ is a symmetric bilinear form on $V$ , so $η (v, w)$ is additive and homogeneous in each argument and $η (v, w) = η (w, v)$ ; in particular polarisation gives $η (v + w, v + w) = η (v, v) + 2 η (v, w) + η (w, w)$ and $η (λ v, μ w) = λ μ η (v, w)$ for real $λ, μ$ .

[F4]

Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1): there is an ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of $V$ with $η (e_{a}, e_{b}) = d i a g (- 1, 1, 1, 1)_{a b}$ .

[F5]

Existence of future-oriented inertial frames at every event with Inertial frames and inertial charts of the affine Minkowski model: in the oriented time-oriented affine Minkowski space there exists, for every event $o$ , an inertial frame with origin $o$ , and an inertial frame's basis is an $η$ -orthonormal ordered basis whose first vector lies in $C^{+}$ ; in particular such a basis of $V$ exists.

[F6]

The timelike set has exactly two convex components, described in orthonormal coordinates: in every $η$ -orthonormal ordered basis whose first vector is timelike, writing $v = v^{0} e_{0} + v_{s}$ with $v_{s} \in S : = s p a n (e_{1}, e_{2}, e_{3})$ and $∣ v_{s} ∣ : = \sqrt{(v^{1})^{2} + (v^{2})^{2} + (v^{3})^{2}}$ , one has $η (v, v) = - (v^{0})^{2} + ∣ v_{s} ∣^{2}$ , $C^{+} = {v : v^{0} > ∣ v_{s} ∣}$ (for the appropriate naming of the two components) and $c l (C^{+}) = {v : v^{0} \geq ∣ v_{s} ∣}$ .

[F7]

The Euclidean spatial structure and cross product of an inertial frame: for an inertial frame, $S = s p a n (e_{1}, e_{2}, e_{3})$ equals $e_{0}^{⊥}$ , the restriction $δ$ of $η$ to $S$ is positive definite, and $∣ w ∣ = \sqrt{δ (w, w)}$ is a norm on $S$ ; in coordinates $δ (v_{s}, w_{s}) = v_{s} \cdot w_{s} = \sum_{i = 1}^{3} v^{i} w^{i}$ .

[F8]

The Cauchy-Schwarz inequality for finite sums: for real numbers $a_{i}, b_{i}$ , $∣ \sum_{i = 1}^{3} a_{i} b_{i} ∣ \leq (\sum_{i = 1}^{3} a_{i}^{2})^{1 / 2} (\sum_{i = 1}^{3} b_{i}^{2})^{1 / 2}$ .

[F9]

Timelike, null, spacelike and causal vectors of a Lorentz vector space and Time orientation, future and past timelike/null/causal vectors, and the closed cones: a vector $v$ is causal exactly when $v \neq 0$ and $η (v, v) \leq 0$ (it is timelike when $η (v, v) < 0$ and null when $v \neq 0$ and $η (v, v) = 0$ ), and $C^{+}$ is a component of the timelike set with closed cone $c l (C^{+})$ .

[F10]

Rational powers $a^{r}$ of a positive base and Laws of rational exponents: for $a, b > 0$ and rationals $r, s$ the power $a^{r}$ is the positive real number defined there, with $(a b)^{r} = a^{r} b^{r}$ , $(a^{r})^{s} = a^{r s}$ and $a^{1 / 2}$ the unique nonnegative square root of $a$ ; in particular $\sqrt{λ^{2}} = λ$ for $λ > 0$ and $γ \sqrt{1 - γ^{- 2}} = \sqrt{γ^{2} - 1}$ for $γ \geq 1$ .

Proof

technique · fix orthonormal coordinates, prove the algebraic identity and positivity there, then specialise to $P=mU$ using the relative-velocity decomposition

1.1F1F2F3

By [F1], $q_{n} (P) = P + η (P, n) n \in n^{⊥}$ , $η (P, n) = - E_{n} (P) / c$ , $m (P)^{2} = - η (P, P) / c^{2}$ for nonzero future causal $P$ , and $T_{n} (P) = E_{n} (P) - m c^{2}$ in the setting of (b). Expanding with [F3] and $η (n, n) = - 1$ gives $η (q_{n} (P), q_{n} (P)) = η (P, P) + 2 η (P, n)^{2} + η (P, n)^{2} η (n, n) = η (P, P) + η (P, n)^{2} = η (P, P) + E_{n} (P)^{2} / c^{2}$ . The formula $m (P)^{2} = - η (P, P) / c^{2}$ involves only the scalar $η (P, P)$ , so $m (P)$ depends on no observer and no basis, whereas $E_{n}$ and $q_{n}$ are built from $η (P, n)$ and $h$ and depend on $n$ .

1.2F3F4F5F6F7F8

Choose by [F4] and [F5] an $η$ -orthonormal ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ with $e_{0} \in C^{+}$ ; write $v = v^{0} e_{0} + v_{s}$ with $v_{s} \in S = s p a n (e_{1}, e_{2}, e_{3})$ and $∣ v_{s} ∣$ as in [F6]. Bilinearity [F3] extends the quadratic form of [F6] to $η (v, w) = - v^{0} w^{0} + δ (v_{s}, w_{s})$ with $δ (v_{s}, w_{s}) = v_{s} \cdot w_{s}$ the Euclidean inner product of [F7]. By [F6], $C^{+} = {v : v^{0} > ∣ v_{s} ∣}$ and $c l (C^{+}) = {v : v^{0} \geq ∣ v_{s} ∣}$ ; hence $n^{0} > ∣ n_{s} ∣ \geq 0$ and $P^{0} \geq ∣ P_{s} ∣ \geq 0$ , while $η (n, n) = - 1$ gives $(n^{0})^{2} = 1 + ∣ n_{s} ∣^{2}$ . By [F8], $∣ P_{s} \cdot n_{s} ∣ \leq ∣ P_{s} ∣ ∣ n_{s} ∣$ .

1.3F2F3F7

In the setting of (b), put $u : = U / c$ . Since $η (U, U) = - c^{2}$ , bilinearity [F3] gives $η (u, u) = c^{- 2} η (U, U) = - 1$ , so $u$ is future unit timelike, and [F2] clause (d) applies: with $γ : = - η (n, u) \geq 1$ and $w : = c h (u) / γ$ one has $w \in n^{⊥}$ , $u = γ (n + w / c)$ , $∣ w ∣^{2} = c^{2} (1 - γ^{- 2})$ , and $γ = 1$ if and only if $u = n$ ; equivalently $U = c n$ , and $P = m U = m c n$ when $γ = 1$ .

2.1step 1.1step 1.2F8

Positivity of $E_{n}$ : using [step 1.2], $c^{- 1} E_{n} (P) = - η (P, n) = P^{0} n^{0} - P_{s} \cdot n_{s}$ , so $c^{- 1} E_{n} (P) \geq P^{0} n^{0} - ∣ P_{s} ∣ ∣ n_{s} ∣ \geq ∣ P_{s} ∣ n^{0} - ∣ P_{s} ∣ ∣ n_{s} ∣ = ∣ P_{s} ∣ (n^{0} - ∣ n_{s} ∣) \geq 0$ , where the first inequality uses [F8], the second uses $P^{0} \geq ∣ P_{s} ∣$ and $n^{0} > 0$ , and the last uses $n^{0} > ∣ n_{s} ∣$ of [step 1.2]; if $∣ P_{s} ∣ > 0$ the last term is strictly positive, while if $∣ P_{s} ∣ = 0$ then $P \neq 0$ forces $P^{0} > 0$ and $c^{- 1} E_{n} (P) = P^{0} n^{0} > 0$ , so $E_{n} (P) > 0$ in both cases.

2.2step 1.1F1F2

Since $q_{n} (P) \in n^{⊥}$ by step 1.1 and the restriction of $η$ to $n^{⊥}$ is positive definite by [F2], $η (q_{n} (P), q_{n} (P)) = ∣ q_{n} (P) ∣^{2} \geq 0$ . Combining this with the identity of step 1.1 gives the displayed equality and inequality of (a).

2.3step 1.1step 1.3F2F3

The energy and momentum of $P = m U$ : by [step 1.1] and [step 1.3], $E_{n} (P) = - c η (m U, n) = - m c^{2} η (U / c, n) = m c^{2} γ$ and $q_{n} (P) = h (m U) = m h (U) = m c h (u) = m γ w$ , the last equality because $h (u) = h (γ (n + w / c)) = γ h (w / c) = γ w / c$ by linearity of $h$ and $h (n) = 0$ in [F2].

3.1step 2.1step 2.2F1F9

The null case: if $P$ is future null then $η (P, P) = 0$ by [F9], so [step 2.2] gives $η (q_{n} (P), q_{n} (P)) = E_{n} (P)^{2} / c^{2} > 0$ by [step 2.1], and since $q_{n} (P) \in n^{⊥}$ the norm of [F1] gives $∣ q_{n} (P) ∣ = \sqrt{η (q_{n} (P), q_{n} (P))} = E_{n} (P) / c$ , that is $E_{n} (P) = c ∣ q_{n} (P) ∣ > 0$ .

3.2step 1.1step 2.2F1

The timelike case and the mass shell: if $P$ is future timelike, then $η (P, P) = - m (P)^{2} c^{2}$ by the definition of $m (P)$ in [F1]; substituting this into [step 2.2] and multiplying by $c^{2}$ gives $E_{n} (P)^{2} = c^{2} η (q_{n} (P), q_{n} (P)) + m (P)^{2} c^{4} = c^{2} ∣ q_{n} (P) ∣^{2} + m (P)^{2} c^{4}$ , using $∣ q_{n} (P) ∣^{2} = η (q_{n} (P), q_{n} (P))$ for $q_{n} (P) \in n^{⊥}$ .

3.3step 1.3step 2.3F1

The kinetic energy: by [step 2.3] and $T_{n} (P) = E_{n} (P) - m c^{2}$ of [F1], $T_{n} (P) = m c^{2} γ - m c^{2} = m c^{2} (γ - 1) \geq 0$ because $γ \geq 1$ in [step 1.3]; moreover $T_{n} (P) = 0$ if and only if $γ = 1$ , which by [step 1.3] is equivalent to $u = n$ , that is to $U = c n$ , that is to $P = m U = m c n$ , and conversely $P = m c n$ gives $γ = 1$ and $T_{n} (P) = 0$ .

3.4step 1.3step 2.3F10

The momentum magnitude: by [step 2.3] and [step 1.3], $∣ q_{n} (P) ∣ = m γ ∣ w ∣ = m γ c \sqrt{1 - γ^{- 2}}$ , and since $γ \geq 1$ , [F10] gives $γ \sqrt{1 - γ^{- 2}} = \sqrt{γ^{2} (1 - γ^{- 2})} = \sqrt{γ^{2} - 1}$ , so $∣ q_{n} (P) ∣ = m c \sqrt{γ^{2} - 1}$ .

4.1step 1.1step 2.1step 2.3step 3.1step 3.2step 3.3step 3.4F3∎

Collect the conclusions and evaluate the invariant mass of $P = m U$ : by [step 1.1], $m (P)^{2} = - η (P, P) / c^{2}$ , and for $P = m U$ with $η (U, U) = - c^{2}$ bilinearity [F3] gives $η (P, P) = m^{2} η (U, U) = - m^{2} c^{2}$ , so $m (P)^{2} = m^{2}$ and, since both $m (P)$ and $m$ are nonnegative and $c > 0$ , $m (P) = m$ ; together with [step 2.1] and [step 3.1] this proves clause (a) including (i), with [step 3.2] proving clause (a)(ii), and with [step 2.3], [step 3.3] and [step 3.4] it proves clause (b), the vanishing statement of (b) included by [step 3.3].

Positivity, mass shell and kinetic energy of the observer quantities of a four-momentum

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources