Relativistic screw: Difference between revisions
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Eric Lengyel (talk | contribs) (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$$ about a line $$\boldsymbol l$$ is given by | ||
:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta | :$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ , | ||
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 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 | ||
Revision as of 10:03, 24 November 2024
A relativistic screw $$\mathbf Q$$ about a 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 operator $$\mathbf Q$$ transforms a position $$\mathbf r$$ (or any other quantity) through the sandwich product
- $$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$.