{"batchcomplete":"","continue":{"lecontinue":"20240725072412|1","continue":"-||"},"query":{"logevents":[{"logid":11,"ns":6,"title":"File:Antimetric-pstga.svg","pageid":9,"logpage":9,"revid":19,"params":{},"type":"create","action":"create","user":"Eric Lengyel","timestamp":"2024-11-25T08:53:36Z","comment":""},{"logid":10,"ns":6,"title":"File:Antimetric-pstga.svg","pageid":9,"logpage":9,"revid":19,"params":{"img_sha1":"8jddbvl3xwdtxr9zdb3hx009mluvjqc","img_timestamp":"2024-11-25T08:53:36Z"},"type":"upload","action":"upload","user":"Eric Lengyel","timestamp":"2024-11-25T08:53:36Z","comment":""},{"logid":9,"ns":6,"title":"File:Metric-pstga.svg","pageid":8,"logpage":8,"revid":18,"params":{},"type":"create","action":"create","user":"Eric Lengyel","timestamp":"2024-11-25T08:53:23Z","comment":""},{"logid":8,"ns":6,"title":"File:Metric-pstga.svg","pageid":8,"logpage":8,"revid":18,"params":{"img_sha1":"m91xws35t20w2m7iej13o1u6z1qfhx8","img_timestamp":"2024-11-25T08:53:23Z"},"type":"upload","action":"upload","user":"Eric Lengyel","timestamp":"2024-11-25T08:53:23Z","comment":""},{"logid":7,"ns":0,"title":"Metrics","pageid":7,"logpage":7,"revid":17,"params":{},"type":"create","action":"create","user":"Eric Lengyel","timestamp":"2024-11-25T08:50:46Z","comment":"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...\""},{"logid":6,"ns":0,"title":"Relativistic screw","pageid":6,"logpage":6,"revid":13,"params":{},"type":"create","action":"create","user":"Eric Lengyel","timestamp":"2024-11-24T09:56:56Z","comment":"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  :$$\\mathbf r' = \\mathbf Q \\mathbin{\\unicode{x27C7}}...\""},{"logid":5,"ns":0,"title":"Translation","pageid":5,"logpage":5,"revid":12,"params":{},"type":"create","action":"create","user":"Eric Lengyel","timestamp":"2024-11-20T01:45:50Z","comment":"Created page with \"A spacetime translation operator $$\\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...\""},{"logid":4,"ns":0,"title":"Velocity","pageid":4,"logpage":4,"revid":10,"params":{},"type":"create","action":"create","user":"Eric Lengyel","timestamp":"2024-11-20T01:24:07Z","comment":"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]]\""},{"logid":3,"ns":0,"title":"Position","pageid":3,"logpage":3,"revid":8,"params":{},"type":"create","action":"create","user":"Eric Lengyel","timestamp":"2024-11-20T01:20:57Z","comment":"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.\""},{"logid":2,"ns":0,"title":"Momentum","pageid":2,"logpage":2,"revid":3,"params":{},"type":"create","action":"create","user":"Eric Lengyel","timestamp":"2024-11-10T23:49:23Z","comment":"Created page with \"The momentum $$\\mathbf P$$ is a bivector quantity with the following ten components.  :$$\\mathbf P = m\\mathbf r \\wedge \\dfrac{d\\mathbf r}{d\\tau} = \\gamma m \\left[c \\mathbf e_{40} + \\dot x \\mathbf e_{41} + \\dot y \\mathbf e_{42} + \\dot z \\mathbf e_{43} + (y\\dot z - z\\dot y) \\mathbf e_{23} + (z\\dot x - x\\dot z) \\mathbf e_{31} + (x\\dot y - y\\dot x) \\mathbf e_{12} + c(x - \\dot x t) \\mathbf e_{10} + c(y - \\dot y t) \\mathbf e_{20} + c(z - \\dot z t) \\mathbf e_{30} \\right]$$  ==...\""}]}}