Relativistic screw: Difference between revisions

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(Created page with "A relativistic screw $$\mathbf Q$$ is given by :$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta\,\mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ , where $$c$$ is the speed of light and $$\gamma = dt/d\tau$$. The operator $$\mathbf Q$$ transforms a position $$\mathbf r$$ (or any other quantity) through the sandwich product :$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}}...")
 
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A relativistic screw $$\mathbf Q$$ is given by
A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by


:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta\,\mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,
:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,


where $$c$$ is the speed of light and $$\gamma = dt/d\tau$$.
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
 
:$$\mathbf Q(2\tau) = \boldsymbol l\sin(\gamma\tau\dot\phi) - (\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0)\cos(\gamma\tau\dot\phi) - \gamma\tau\dot\delta \mathbf e_0 \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi) - [\boldsymbol l \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi)] \vee \gamma c\tau\,\mathbf e_{321}$$.


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


:$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$.
:$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$.
The bulk [[norm]] of a unitized relativistic screw operator is given by
:$$\left\Vert\mathbf Q(2\tau)\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{c^2t^2 - Q_{mx}^2 - Q_{my}^2 - Q_{mz}^2 - Q_{mw}^2}$$ ,
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).


== See Also ==
== See Also ==


* [[Translation]]
* [[Translation]]

Latest revision as of 01:54, 21 December 2024

A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by

$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,

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

$$\mathbf Q(2\tau) = \boldsymbol l\sin(\gamma\tau\dot\phi) - (\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0)\cos(\gamma\tau\dot\phi) - \gamma\tau\dot\delta \mathbf e_0 \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi) - [\boldsymbol l \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi)] \vee \gamma c\tau\,\mathbf e_{321}$$.

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

$$\left\Vert\mathbf Q(2\tau)\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{c^2t^2 - Q_{mx}^2 - Q_{my}^2 - Q_{mz}^2 - Q_{mw}^2}$$ ,

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).

See Also