Relativistic screw: Difference between revisions
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A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by | A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by | ||
:$$\mathbf Q( | :$$\mathbf Q(\tau) = \exp_\unicode{x27C7}\left[\dfrac{1}{2}\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \frac{1}{2}\gamma c\tau\,\mathbf e_{321}\right]$$ , | ||
where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to | where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to | ||
:$$\mathbf Q( | :$$\mathbf Q(\tau) = \boldsymbol l\sin\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) + {\large\unicode{x1D7D9}}\cos\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) - \left(\dfrac{1}{2}\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0\right)\cos\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) - \dfrac{1}{2}\gamma\tau\dot\delta \mathbf e_0 \sin\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) - \dfrac{1}{2}\gamma c\tau\left[\boldsymbol l \sin\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) + {\large\unicode{x1D7D9}}\cos\left(\dfrac{1}{2}\gamma\tau\dot\phi\right)\right] \vee \mathbf e_{321}$$. | ||
This operator has 12 components and can be written generically as | |||
:$$\mathbf Q(\tau) = q_x\,\mathbf e_{410} + q_y\,\mathbf e_{420} + q_z\,\mathbf e_{430} + q_w\,\unicode{x1D7D9} + m_x\,\mathbf e_{230} + m_y\,\mathbf e_{310} + m_z\,\mathbf e_{120} + m_w\,\mathbf e_0 + s_x\,\mathbf e_1 + s_y\,\mathbf e_2 + s_z\,\mathbf e_3 + s_w\,e_{321}$$ . | |||
The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product | The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product | ||
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The bulk [[norm]] of a unitized relativistic screw operator is given by | The bulk [[norm]] of a unitized relativistic screw operator is given by | ||
:$$\left\Vert\mathbf Q( | :$$\left\Vert\mathbf Q(\tau)\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \dfrac{1}{2}\sqrt{\gamma^2c^2\tau^2 - m_x^2 - m_y^2 - m_z^2 - m_w^2}$$ , | ||
and it corresponds to the distance that the origin has moved after time $$\tau$$. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light). | and it corresponds to the distance that the origin has moved after time $$\tau$$. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light). | ||
Latest revision as of 08:55, 26 December 2024
A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by
- $$\mathbf Q(\tau) = \exp_\unicode{x27C7}\left[\dfrac{1}{2}\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \frac{1}{2}\gamma c\tau\,\mathbf e_{321}\right]$$ ,
where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to
- $$\mathbf Q(\tau) = \boldsymbol l\sin\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) + {\large\unicode{x1D7D9}}\cos\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) - \left(\dfrac{1}{2}\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0\right)\cos\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) - \dfrac{1}{2}\gamma\tau\dot\delta \mathbf e_0 \sin\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) - \dfrac{1}{2}\gamma c\tau\left[\boldsymbol l \sin\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) + {\large\unicode{x1D7D9}}\cos\left(\dfrac{1}{2}\gamma\tau\dot\phi\right)\right] \vee \mathbf e_{321}$$.
This operator has 12 components and can be written generically as
- $$\mathbf Q(\tau) = q_x\,\mathbf e_{410} + q_y\,\mathbf e_{420} + q_z\,\mathbf e_{430} + q_w\,\unicode{x1D7D9} + m_x\,\mathbf e_{230} + m_y\,\mathbf e_{310} + m_z\,\mathbf e_{120} + m_w\,\mathbf e_0 + s_x\,\mathbf e_1 + s_y\,\mathbf e_2 + s_z\,\mathbf e_3 + s_w\,e_{321}$$ .
The operator $$\mathbf Q$$ transforms a position $$\mathbf r$$ (or any other quantity) through the sandwich product
- $$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$.
The bulk norm of a unitized relativistic screw operator is given by
- $$\left\Vert\mathbf Q(\tau)\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \dfrac{1}{2}\sqrt{\gamma^2c^2\tau^2 - m_x^2 - m_y^2 - m_z^2 - m_w^2}$$ ,
and it corresponds to the distance that the origin has moved after time $$\tau$$. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).