Translation - Revision history

Eric Lengyel at 08:32, 26 December 2024

2024-12-26T08:32:27Z

← Older revision		Revision as of 08:32, 26 December 2024
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	A spacetime translation operator $$\mathbf T$$ is given by		A spacetime translation operator $$\mathbf T$$ is given by

	:$$\mathbf T(\tau) = \dfrac{1}{2}\gamma\tau\,(\dot x\,\mathbf e_{230} + \dot y\,\mathbf e_{310} + \dot z\,\mathbf e_{120~~} - c\,\mathbf e_{321~~}) + {\large\unicode{x1D7D9}}$$ .		:$$\mathbf T(\tau) = \dfrac{1}{2}\gamma\tau\,(-c\,\mathbf e_{321} + \dot x\,\mathbf e_{230} + \dot y\,\mathbf e_{310} + \dot z\,\mathbf e_{120}) + {\large\unicode{x1D7D9}}$$ .

	It transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product		The trivector part of this operator is $$\frac{1}{2}\tau (\mathbf e_4 \wedge \mathbf u)^\unicode{x2606}$$ for a [[velocity]] $$\mathbf u$$, where the dualization applies the metric and causes the temporal component to be negated.

			The operator $$\mathbf T(\tau)$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product

	:$$\mathbf r' = \mathbf T \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}}$$.		:$$\mathbf r' = \mathbf T \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}}$$.
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	The bulk [[norm]] of a translation operator is given by		The bulk [[norm]] of a translation operator is given by

	:$$\left\Vert\mathbf T\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{c~~^2t~~^2 - x^2 - y^2 - z^2}$$ ,		:$$\left\Vert\mathbf T\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \dfrac{1}{2}\gamma\tau \sqrt{c^2 - \dot x^2 - \dot y^2 - \dot z^2}$$ ,

			and this makes it clear that the bulk norm is real precisely when the velocity of the translation must be less than the speed of light. Setting $$dt = \gamma\tau$$, $$dx = \dot x\,dt$$, $$dy = \dot y\,dt$$, and $$dz = \dot z\,dt$$, we can rewrite this as

			:$$\left\Vert\mathbf T\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \dfrac{1}{2}\sqrt{c^2 dt^2 - dx^2 - dy^2 - dz^2}$$ .

	~~and it must be real~~ for ~~any motion that's physically possible (i.e., without exceeding the speed of light)~~.		This spacetime interval is a Lorentz invariant having the same value for all inertial observers.

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

Eric Lengyel at 01:22, 21 December 2024

2024-12-21T01:22:55Z

← Older revision		Revision as of 01:22, 21 December 2024
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	A spacetime translation operator $$\mathbf T$$ is given by		A spacetime translation operator $$\mathbf T$$ is given by

	:$$\mathbf T(2\tau) = \gamma \dot x~~\tau~~\,\mathbf e_{230} + ~~\gamma~~ \dot y~~\tau~~\,\mathbf e_{310} + ~~\gamma~~ \dot z~~\tau~~\,\mathbf e_{120} - ~~\gamma~~ c~~\tau~~\,\mathbf e_{321} + {\large\unicode{x1D7D9}}$$ .		:$$\mathbf T(\tau) = \dfrac{1}{2}\gamma\tau\,(\dot x\,\mathbf e_{230} + \dot y\,\mathbf e_{310} + \dot z\,\mathbf e_{120} - c\,\mathbf e_{321}) + {\large\unicode{x1D7D9}}$$ .

	It transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product		It transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product

Eric Lengyel at 10:03, 24 November 2024

2024-11-24T10:03:30Z

← Older revision		Revision as of 10:03, 24 November 2024
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	* [[Velocity]]		* [[Velocity]]
			* [[Relativistic screw]]

Eric Lengyel: Created page with "A spacetime translation operator $$\mathbf T$$ is given by :$$\mathbf T(2\tau) = \gamma \dot x\tau\,\mathbf e_{230} + \gamma \dot y\tau\,\mathbf e_{310} + \gamma \dot z\tau\,\mathbf e_{120} - \gamma c\tau\,\mathbf e_{321} + {\large\unicode{x1D7D9}}$$ . It transforms a position $$\mathbf r$$ (or any other quantity) through the sandwich product :$$\mathbf r' = \mathbf T \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\uni..."

2024-11-20T01:45:50Z

Created page with "A spacetime translation operator $$\mathbf T$$ is given by :$$\mathbf T(2\tau) = \gamma \dot x\tau\,\mathbf e_{230} + \gamma \dot y\tau\,\mathbf e_{310} + \gamma \dot z\tau\,\mathbf e_{120} - \gamma c\tau\,\mathbf e_{321} + {\large\unicode{x1D7D9}}$$ . It transforms a position $$\mathbf r$$ (or any other quantity) through the sandwich product :$$\mathbf r' = \mathbf T \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\uni..."

New page

A spacetime translation operator $$\mathbf T$$ is given by

:$$\mathbf T(2\tau) = \gamma \dot x\tau\,\mathbf e_{230} + \gamma \dot y\tau\,\mathbf e_{310} + \gamma \dot z\tau\,\mathbf e_{120} - \gamma c\tau\,\mathbf e_{321} + {\large\unicode{x1D7D9}}$$ .

It transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product

:$$\mathbf r' = \mathbf T \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}}$$.

The bulk [[norm]] of a translation operator is given by

:$$\left\Vert\mathbf T\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{c^2t^2 - x^2 - y^2 - z^2}$$ ,

and it must be real for any motion that's physically possible (i.e., without exceeding the speed of light).

== See Also ==

* [[Velocity]]