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- 08:50, 25 November 2024 Metrics (hist | edit) [757 bytes] Eric Lengyel (talk | contribs) (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...")
- 09:56, 24 November 2024 Relativistic screw (hist | edit) [1,656 bytes] 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 :$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}}...")
- 01:45, 20 November 2024 Translation (hist | edit) [805 bytes] Eric Lengyel (talk | contribs) (Created page with "A spacetime translation operator $$\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...")
- 01:24, 20 November 2024 Velocity (hist | edit) [397 bytes] Eric Lengyel (talk | contribs) (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")
- 01:20, 20 November 2024 Position (hist | edit) [240 bytes] Eric Lengyel (talk | contribs) (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.")
- 23:49, 10 November 2024 Momentum (hist | edit) [1,311 bytes] Eric Lengyel (talk | contribs) (Created page with "The momentum $$\mathbf P$$ is a bivector quantity with the following ten components. :$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ ==...")