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Worked example: four-velocity and four-acceleration of uniform circular motion

Example

Worked computation. In an inertial chart with $η_{0} = diag (- 1, 1, 1, 1)$ let $c > 0$ , $R > 0$ and let $ω$ be real with $R ∣ ω ∣ < c$ . Here $\sin, \cos : R \to R$ are the everywhere-convergent real power-series functions of Sine and cosine defined by their real power series; their derivatives are $\cos$ and $- \sin$ (The derivatives of sine and cosine are cosine and minus sine), and $\sin^{2} + \cos^{2} = 1$ (Parity and the Pythagorean identity for sine and cosine). For $t \in R$ , consider the coordinate worldline $x (t) = (c t, R \cos (ω t), R \sin (ω t), 0)$ , parametrized by coordinate time: uniform circular motion in the first two spatial coordinates. Then $w (t) = (- R ω \sin (ω t), R ω \cos (ω t), 0)$ has $∣ w ∣ = R ∣ ω ∣$ , so $- η (x^{'}, x^{'}) = c^{2} - R^{2} ω^{2}$ and the proper-time factor is $τ^{'} = c^{- 1} \sqrt{c^{2} - R^{2} ω^{2}} = 1 / γ$ with $γ : = (1 - R^{2} ω^{2} / c^{2})^{- 1 / 2}$ ; the four-velocity is $U = γ (c, w)$ . Since $∣ w ∣$ is constant, $γ$ is constant; the coordinate acceleration is $a (t) = (- R ω^{2} \cos (ω t), - R ω^{2} \sin (ω t), 0)$ , with $∣ a ∣ = R ω^{2}$ and $w \cdot a = 0$ . By clause (e) of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition the four-acceleration is $A = (0, γ^{2} a)$ , so $A^{0} = 0$ , $η (U, A) = 0$ and $η (A, A) = γ^{4} R^{2} ω^{4} \geq 0$ . Its magnitude is $γ^{2} R ω^{2}$ ; for $R > 0$ it is spacelike exactly when $ω \neq 0$ , while $ω = 0$ is the static limit with $γ = 1$ , $U = (c, 0, 0, 0)$ and $A = 0$ . Numerical specialization: if $R ω = 3 c / 5$ , then $γ = 5 / 4$ and the magnitude of the four-acceleration is $γ^{2} R ω^{2} = (25 / 16) R ω^{2} = (15 c ω) / 16$ .

Facts & Assumptions

Given: An inertial chart with $η_{0} = diag (- 1, 1, 1, 1)$ , reals $c > 0$ , $R > 0$ and $ω$ with $R ∣ ω ∣ < c$ , and the worldline $x : R \to A$ with coordinates $x (t) = (c t, R \cos (ω t), R \sin (ω t), 0)$ .

[L1]

Chart and spatial structure (Inertial frames and inertial charts of the affine Minkowski model, The Euclidean spatial structure and cross product of an inertial frame): in the chart $η_{0} = diag (- 1, 1, 1, 1)$ , the spatial part carries the Euclidean norm $∣ \cdot ∣$ and inner product $w \cdot z$ .

[L2]

Curves (Curves, regular causal curves and future timelike worldlines in affine Minkowski space, Timelike, null, spacelike and causal vectors of a Lorentz vector space): $x$ is $C^{1}$ with tangent $x^{'} = (c, w)$ and is a future timelike worldline exactly when $η (x^{'}, x^{'}) < 0$ with $x^{'}$ future.

[L3]

Proper time and four-velocity (Proper time along a future timelike worldline, Four-velocity and four-acceleration of a future timelike worldline): $τ^{'} = c^{- 1} \sqrt{- η (x^{'}, x^{'})}$ , $U = x^{'} / τ^{'}$ , and $A = U^{'} / τ^{'}$ when $x$ is $C^{2}$ .

[L4]

Coordinate form of the decomposition (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition, clause (e)): for a $C^{2}$ future timelike worldline with coordinates $t \mapsto (c t, r (t))$ , $w = r^{'}$ , $a = w^{'}$ , one has $∣ w ∣ < c$ , $U = γ_{w} (c, w)$ with $γ_{w} = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2}$ , $A^{0} = γ_{w}^{4} (w \cdot a) / c$ , $A_{v e c} = γ_{w}^{2} a + γ_{w}^{4} (w \cdot a) w / c^{2}$ , and $η (A, A) = γ_{w}^{4} ∣ a ∣^{2} + γ_{w}^{6} (w \cdot a)^{2} / c^{2}$ , vanishing exactly when $a = 0$ .

[L5]

Bilinearity and the diagonal form (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space): $η$ is bilinear, so $η (U, U)$ is computed from the chart matrix in [L1].

[L6]

Trigonometric calculus: $\sin, \cos$ are defined on all of $R$ by their convergent power series (Sine and cosine defined by their real power series), with $\sin' = \cos$ , $\cos' = - \sin$ and values $\sin 0 = 0$ , $\cos 0 = 1$ (The derivatives of sine and cosine are cosine and minus sine). The identity $\sin^{2} θ + \cos^{2} θ = 1$ holds for every real $θ$ (Parity and the Pythagorean identity for sine and cosine). The chain rule (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ) gives $(\sin (ω t))' = ω \cos (ω t)$ and $(\cos (ω t))' = - ω \sin (ω t)$ ; iterating these identities makes the coordinate path $C^{\infty}$ .

Verification

technique · direct

1.1givenL1L2L6

The difference quotients of the first coordinate give $c$ ; those of the spatial coordinates give $w (t) = (- R ω \sin (ω t), R ω \cos (ω t), 0)$ , so $x^{'} (t) = (c, w (t))$ ; then $∣ w (t) ∣ = R ∣ ω ∣$ and, by [L1], $η (x^{'}, x^{'}) = - c^{2} + R^{2} ω^{2} = - (c^{2} - R^{2} ω^{2}) < 0$ by the hypothesis $R ∣ ω ∣ < c$ , with $x^{'}$ future; hence $x$ is a $C^{1}$ future timelike worldline by [L2].

2.1step 1.1L3

By [L3], $τ^{'} = c^{- 1} \sqrt{c^{2} - R^{2} ω^{2}} = (1 - R^{2} ω^{2} / c^{2})^{1 / 2} = 1 / γ$ with $γ : = (1 - R^{2} ω^{2} / c^{2})^{- 1 / 2}$ , and $U = x^{'} / τ^{'} = γ (c, w)$ .

2.2step 1.1L6

The spatial path is $r (t) = (R \cos (ω t), R \sin (ω t), 0)$ , with $r^{'} = w$ and $r^{''} = a (t) = (- R ω^{2} \cos (ω t), - R ω^{2} \sin (ω t), 0)$ ; hence $∣ a ∣ = R ω^{2}$ and $w \cdot a = R^{2} ω^{3} (\sin \cdot \cos - \cos \cdot \sin) = 0$ .

3.1step 2.1step 2.2L4L5

Since $∣ w ∣ = R ∣ ω ∣$ is constant, $γ$ is constant and $A = γ_{w} \frac{d}{d t} (γ_{w} (c, w))$ with $γ_{w} = γ$ ; clause (e) of [L4] with $w \cdot a = 0$ gives $A = (0, γ^{2} a)$ , so $A^{0} = 0$ , and $η (U, A) = 0$ ; also $∣ a ∣ = R ω^{2}$ gives $η (A, A) = γ^{4} ∣ a ∣^{2} = γ^{4} R^{2} ω^{4} \geq 0$ , with equality if and only if $a = 0$ , i.e. if and only if $ω = 0$ (since $R > 0$ ); the magnitude of the four-acceleration is then $γ^{2} R ω^{2}$ .

4.1step 2.1step 3.1

At $ω = 0$ the worldline is $x (t) = (c t, R, 0, 0)$ , the static limit: $w = 0$ , $γ = 1$ , $U = (c, 0, 0, 0)$ and $A = 0$ , so the clock is at rest in the chart and its acceleration vanishes; for $ω \neq 0$ the four-acceleration is spacelike with $η (A, A) = γ^{4} R^{2} ω^{4} > 0$ .

5.1step 3.1∎

Numerical specialization: if $R ω = 3 c / 5$ , then $γ = (1 - 9 / 25)^{- 1 / 2} = 5 / 4$ and the magnitude of the four-acceleration is $γ^{2} R ω^{2} = (25 / 16) R ω^{2} = (25 / 16) (3 c / 5) ω = (15 c ω) / 16$ , with the dimensions of acceleration because $c ω$ has units m/s $^{2}$ when $ω$ is an angular rate.

Remarks

The zero-frequency case is retained deliberately. At $ω = 0$ the clock is static and its four-acceleration vanishes; the acceleration square $η (A, A) = γ^{4} R^{2} ω^{4}$ is nonnegative always and positive exactly for nonzero $ω$ when $R > 0$ .
Centripetal acceleration is orthogonal to the velocity. The identity $w \cdot a = 0$ is what forces $A^{0} = 0$ and keeps the four-acceleration purely spatial; it reflects the constant speed of the motion and is not an additional dynamical assumption.

Worked example: four-velocity and four-acceleration of uniform circular motion

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels