Spacetime Geometric Algebra - User contributions [en]

Relativistic screw

2025-02-08T06:33:35Z

Eric Lengyel:

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

:$$\mathbf Q(\tau) = \exp_\unicode{x27C7}\left[\dfrac{1}{2}\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \dfrac{1}{2}\gamma c\tau\,\mathbf e_{321}\right]$$ ,

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).

== See Also ==

* [[Translation]]

Relativistic screw

2024-12-26T08:55:00Z

Eric Lengyel:

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).

== See Also ==

* [[Translation]]

Translation

2024-12-26T08:32:27Z

Eric Lengyel:

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

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

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

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

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

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

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

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

This spacetime interval is a Lorentz invariant having the same value for all inertial observers.

== See Also ==

* [[Velocity]]
* [[Relativistic screw]]

Relativistic screw

2024-12-21T01:54:57Z

Eric Lengyel:

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 ==

* [[Translation]]

Relativistic screw

2024-12-21T01:52:25Z

Eric Lengyel:

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

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

where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The half rate of rotation about the line is specified by $$\dot\phi$$, and the half rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to

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

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

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

and it corresponds to the distance that the origin is moved. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).

== See Also ==

* [[Translation]]

Translation

2024-12-21T01:22:55Z

Eric Lengyel:

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

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

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]]
* [[Relativistic screw]]

Relativistic screw

2024-12-09T08:20:52Z

Eric Lengyel:

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

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

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

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

and it corresponds to the distance that the origin is moved. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).

== See Also ==

* [[Translation]]

Relativistic screw

2024-12-09T06:36:47Z

Eric Lengyel:

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 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

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

and it corresponds to the distance that the origin is moved. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).

== See Also ==

* [[Translation]]

File:Antimetric-pstga.svg

2024-11-25T08:53:36Z

Eric Lengyel:

File:Metric-pstga.svg

2024-11-25T08:53:23Z

Eric Lengyel:

Metrics

2024-11-25T08:50:46Z

Eric Lengyel: Created page with "The ''metric'' used in the 5D projective geometric algebra over 4D spacetime is the $$5 \times 5$$ matrix $$\mathfrak g$$ given by :$$\mathfrak g = \begin{bmatrix} -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\\end{bmatrix}$$ . The ''metric exomorphism matrix'' $$\mathbf G$$, often just called the "metric" itself, corresponding to the metric $$\mathfrak g$$ is the $$32 \times 32$$ matrix shown below. Image:m..."

The ''metric'' used in the 5D projective geometric algebra over 4D spacetime is the $$5 \times 5$$ matrix $$\mathfrak g$$ given by

:$$\mathfrak g = \begin{bmatrix}
-1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 \\\end{bmatrix}$$ .

The ''metric exomorphism matrix'' $$\mathbf G$$, often just called the "metric" itself, corresponding to the metric $$\mathfrak g$$ is the $$32 \times 32$$ matrix shown below.

[[Image:metric-pstga.svg|800px]]

The ''metric antiexomorphism matrix'' $$\mathbb G$$, often called the "antimetric", corresponding to the metric $$\mathfrak g$$ is the $$32 \times 32$$ matrix shown below.

[[Image:antimetric-pstga.svg|800px]]

== See Also ==

* [[Duals]]
* [[Dot products]]

Relativistic screw

2024-11-24T10:08:40Z

Eric Lengyel:

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

:$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$.

The bulk [[norm]] of a unitized relativistic screw operator is given by

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

and it corresponds to the distance that the origin is moved. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).

== See Also ==

* [[Translation]]

Translation

2024-11-24T10:03:30Z

Eric Lengyel:

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]]
* [[Relativistic screw]]

Relativistic screw

2024-11-24T10:03:05Z

Eric Lengyel:

Relativistic screw

2024-11-24T09:56:56Z

Eric Lengyel: 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}}..."

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

:$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$.

== See Also ==

* [[Translation]]

Translation

2024-11-20T01:45:50Z

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..."

Velocity

2024-11-20T01:24:58Z

Eric Lengyel:

The velocity $$\mathbf u$$ of a body in spacetime is given by

:$$\mathbf u = \dfrac{d\mathbf r}{d\tau} = \gamma c\,\mathbf e_0 + \gamma \dot x\,\mathbf e_1 + \gamma \dot y\,\mathbf e_2 + \gamma \dot z\,\mathbf e_3$$ ,

where $$\mathbf r$$ is the body's [[position]] and $$\tau$$ is proper time. The velocity $$\mathbf u$$ has units of length / time.

== See Also ==

* [[Position]]
* [[Momentum]]

Velocity

2024-11-20T01:24:07Z

Eric Lengyel: Created page with "The velocity $$\mathbf u$$ of a body in spacetime is given by :$$\mathbf u = \dfrac{d\mathbf r}{d\tau} = \gamma c\,\mathbf e_0 + \gamma \dot x\,\mathbf e_1 + \gamma \dot y\,\mathbf e_2 + \gamma \dot z\,\mathbf e_3$$ , where $$\mathbf r$$ is the body's position and $$\tau$$ is proper time. == See Also == * Position * Momentum"

Position

2024-11-20T01:21:20Z

Eric Lengyel:

The spacetime position $$\mathbf r$$ of a body is given by

:$$\mathbf r = ct\,\mathbf e_0 + x\,\mathbf e_1 + y\,\mathbf e_2 + z\,\mathbf e_3 + \mathbf e_4$$ .

The position $$\mathbf r$$ has units of length.

== See Also ==

* [[Velocity]]

Position

2024-11-20T01:20:57Z

Eric Lengyel: Created page with "The spacetime position $$\mathbf r$$ of a body is given by :$$\mathbf r = ct\,\mathbf e_0 + x\,\mathbf e_1 + y\,\mathbf e_2 + z\,\mathbf e_3 + \mathbf e_4$$ . The position $$\mathbf r$$ has units of length."

Momentum

2024-11-20T01:12:20Z

Eric Lengyel:

The momentum $$\mathbf P$$ is a bivector quantity with the following ten components.

:$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$

The inertia tensor $$\mathcal I$$ is given by

:$$\mathcal I = {\Large\sum} \gamma m{\left[\begin{array}{cccc|ccc|ccc}
-1 & 0 & 0 & 0 & 0 & 0 & 0 & -x & -y & -z \\
0 & 1 & 0 & 0 & 0 & z & -y & -ct & 0 & 0 \\
0 & 0 & 1 & 0 & -z & 0 & x & 0 & -ct & 0 \\
0 & 0 & 0 & 1 & y & -x & 0 & 0 & 0 & -ct \\
\hline
0 & 0 & -z & y & y^2 + z^2 & -xy & -zx & 0 & ctz & -cty \\
0 & z & 0 & -x & -xy & z^2 + x^2 & -yz & -ctz & 0 & ctx \\
0 & -y & x & 0 & -zx & -yz & x^2 + y^2 & cty & -ctx & 0 \\
\hline
-x & -ct & 0 & 0 & 0 & -ctz & cty & c^2t^2 - x^2 & -xy & -zx \\
-y & 0 & -ct & 0 & ctz & 0 & -ctx & -xy & c^2t^2 - y^2 & -yz \\
-z & 0 & 0 & -ct & -cty & ctx & 0 & -zx & -yz & c^2t^2 - z^2
\end{array}\right]}$$ .

As in classical mechanics, the inertia tensor $$\mathcal I$$ has units of mass × length<sup>2</sup>.

== See Also ==

* [[Velocity]]

Momentum

2024-11-13T23:41:40Z

Eric Lengyel:

The momentum $$\mathbf P$$ is a bivector quantity with the following ten components.

:$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$

The inertia tensor $$\mathcal I$$ is given by

:$$\mathcal I = \gamma \sum{\left[\begin{array}{cccc|ccc|ccc}
-m & 0 & 0 & 0 & 0 & 0 & 0 & -mx & -my & -mz \\
0 & m & 0 & 0 & 0 & mz & -my & -mct & 0 & 0 \\
0 & 0 & m & 0 & -mz & 0 & mx & 0 & -mct & 0 \\
0 & 0 & 0 & m & my & -mx & 0 & 0 & 0 & -mct \\
\hline
0 & 0 & -mz & my & m(y^2 + z^2) & -mxy & -mzx & 0 & mctz & -mcty \\
0 & mz & 0 & -mx & -mxy & m(z^2 + x^2) & -myz & -mctz & 0 & mctx \\
0 & -my & mx & 0 & -mzx & -myz & m(x^2 + y^2) & mcty & -mctx & 0 \\
\hline
-mx & -mct & 0 & 0 & 0 & -mctz & mcty & m(c^2t^2 - x^2) & -mxy & -mzx \\
-my & 0 & -mct & 0 & mctz & 0 & -mctx & -mxy & m(c^2t^2 - y^2) & -myz \\
-mz & 0 & 0 & -mct & -mcty & mctx & 0 & -mzx & -myz & m(c^2t^2 - z^2)
\end{array}\right]}$$ .

== See Also ==

* [[Velocity]]

Momentum

2024-11-11T07:17:21Z

Eric Lengyel:

The momentum $$\mathbf P$$ is a bivector quantity with the following ten components.

:$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$

The inertia tensor $$\mathcal I$$ is given by

:$$\mathcal I = \gamma \sum {{\left[\begin{array}{cccc|ccc|ccc}
-m & 0 & 0 & 0 & 0 & 0 & 0 & -mx & -my & -mz \\
0 & m & 0 & 0 & 0 & mz & -my & -mct & 0 & 0 \\
0 & 0 & m & 0 & -mz & 0 & mx & 0 & -mct & 0 \\
0 & 0 & 0 & m & my & -mx & 0 & 0 & 0 & -mct \\
\hline
0 & 0 & -mz & my & m(y^2 + z^2) & -mxy & -mzx & 0 & mctz & -mcty \\
0 & mz & 0 & -mx & -mxy & m(z^2 + x^2) & -myz & -mctz & 0 & mctx \\
0 & -my & mx & 0 & -mzx & -myz & m(x^2 + y^2) & mcty & -mctx & 0 \\
\hline
-mx & -mct & 0 & 0 & 0 & -mctz & mcty & m(c^2t^2 - x^2) & -mxy & -mzx \\
-my & 0 & -mct & 0 & mctz & 0 & -mctx & -mxy & m(c^2t^2 - y^2) & -myz \\
-mz & 0 & 0 & -mct & -mcty & mctx & 0 & -mzx & -myz & m(c^2t^2 - z^2)
\end{array}\right]}}$$ .

== See Also ==

* [[Velocity]]

Momentum

2024-11-11T07:13:07Z

Eric Lengyel:

The momentum $$\mathbf P$$ is a bivector quantity with the following ten components.

:$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$

The inertia tensor $$\mathcal I$$ is given by

:$$\mathcal I = \gamma \sum {{\begin{bmatrix}
-m & 0 & 0 & 0 & 0 & 0 & 0 & -mx & -my & -mz \\
0 & m & 0 & 0 & 0 & mz & -my & -mct & 0 & 0 \\
0 & 0 & m & 0 & -mz & 0 & mx & 0 & -mct & 0 \\
0 & 0 & 0 & m & my & -mx & 0 & 0 & 0 & -mct \\
0 & 0 & -mz & my & m(y^2 + z^2) & -mxy & -mzx & 0 & mctz & -mcty \\
0 & mz & 0 & -mx & -mxy & m(z^2 + x^2) & -myz & -mctz & 0 & mctx \\
0 & -my & mx & 0 & -mzx & -myz & m(x^2 + y^2) & mcty & -mctx & 0 \\
-mx & -mct & 0 & 0 & 0 & -mctz & mcty & m(c^2t^2 - x^2) & -mxy & -mzx \\
-my & 0 & -mct & 0 & mctz & 0 & -mctx & -mxy & m(c^2t^2 - y^2) & -myz \\
-mz & 0 & 0 & -mct & -mcty & mctx & 0 & -mzx & -myz & m(c^2t^2 - z^2)
\end{bmatrix}}}$$ .

== See Also ==

* [[Velocity]]

Momentum

2024-11-10T23:49:23Z

Eric Lengyel: Created page with "The momentum $$\mathbf P$$ is a bivector quantity with the following ten components. :$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ ==..."

Main Page

2024-07-25T07:26:14Z

Eric Lengyel: Replaced content with "Spacetime Geometric Algebra"

Spacetime Geometric Algebra