User contributions for Eric Lengyel

08:5308:53, 25 November 2024 diff hist 0‎ N File:Antimetric-pstga.svg ‎No edit summary current
08:5308:53, 25 November 2024 diff hist 0‎ N File:Metric-pstga.svg ‎No edit summary current
08:5008:50, 25 November 2024 diff hist +757‎ N Metrics ‎ 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..." current

10:0810:08, 24 November 2024 diff hist +375‎ Relativistic screw ‎No edit summary
10:0310:03, 24 November 2024 diff hist +25‎ Translation ‎No edit summary
10:0310:03, 24 November 2024 diff hist +197‎ Relativistic screw ‎No edit summary
09:5609:56, 24 November 2024 diff hist +599‎ N Relativistic screw ‎ 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}}..."

01:4501:45, 20 November 2024 diff hist +799‎ N Translation ‎ 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 :$$\mathbf r' = \mathbf T \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\uni..."
01:2401:24, 20 November 2024 diff hist +55‎ Velocity ‎No edit summary current
01:2401:24, 20 November 2024 diff hist +342‎ N Velocity ‎ Created page with "The velocity $$\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$$ is proper time. == See Also == * Position * Momentum"
01:2101:21, 20 November 2024 diff hist +32‎ Position ‎No edit summary current
01:2001:20, 20 November 2024 diff hist +208‎ N Position ‎ Created page with "The spacetime position $$\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."
01:1201:12, 20 November 2024 diff hist +52‎ Momentum ‎No edit summary current

23:4923:49, 10 November 2024 diff hist +502‎ N Momentum ‎ Created page with "The momentum $$\mathbf P$$ is a bivector quantity with the following ten components. :$$\mathbf P = m\mathbf r \wedge \dfrac{d\mathbf r}{d\tau} = \gamma m \left[c \mathbf e_{40} + \dot x \mathbf e_{41} + \dot y \mathbf e_{42} + \dot z \mathbf e_{43} + (y\dot z - z\dot y) \mathbf e_{23} + (z\dot x - x\dot z) \mathbf e_{31} + (x\dot y - y\dot x) \mathbf e_{12} + c(x - \dot x t) \mathbf e_{10} + c(y - \dot y t) \mathbf e_{20} + c(z - \dot z t) \mathbf e_{30} \right]$$ ==..."

07:2607:26, 25 July 2024 diff hist −728‎ Main Page ‎ Replaced content with "Spacetime Geometric Algebra" current Tag: Replaced

Navigation menu