31:T5ad,wormholeFrame = { return (t) => svg`` }32:[{"id":"9badbb3ed24bab8a12d07e93bfef4299e0119ef57b435f7ce2fcc6d99b7bc720f91eff2021d72036b7d89417c7d619b465ee12daad440e0e36d95437fd505456","href":"https://static.observableusercontent.com/files/9badbb3ed24bab8a12d07e93bfef4299e0119ef57b435f7ce2fcc6d99b7bc720f91eff2021d72036b7d89417c7d619b465ee12daad440e0e36d95437fd505456","name":"image.png","type":"image/png","lastModified":1614142338392.983,"size":179121}] 2f:{"title":"INVERTED UNIVERSE","cells":[{"id":0,"value":"md`# INVERTED UNIVERSE\nLorentzian Wormholes: _a throat joining two asymptotically flat regions_ [1]\n![](${await FileAttachment(\"image.png\").url()})\n\n### References\n[1] [_\"On the Construction and Traversability of Lorentzian Wormholes\"_](https://uu.diva-portal.org/smash/get/diva2:1333284/FULLTEXT01.pdf), Maximilian Svensson\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":922,"value":"md`# Displacement\n\nLet one side be x = +1 and the other x = -1. The width of the throat be _w_. The contours of the wormhole are _u_. At u = 0 we are in the throat. When _u_ is _+ve_ infinity, _x_ is 1 and _r_ is infinity.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1238,"value":"md`## Contours of wormhole are _theta_ and _u_\n\nLets figure out equations describing 3d coordinate given _u_ and _theta_. Theta is simple, so lets concentrate on the profile bases on _u_ first.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1243,"value":"md`Step *u_steps* times from *-u_max* to *+u_max*:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":927,"value":"u_in = range(c.u_steps).map(x => u_step * x - c.u_max)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1221,"value":"u_step = (c.u_max * 2.0) / (c.u_steps-1) ","pinned":false,"hidden":false,"mode":"ojs"},{"id":1115,"value":"md`The radius should rapidly increases as we get further from u = 0\n\nI can't figure out the equations in the Physics papers so I will just use two exponetials so away from zero the value is really high.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":935,"value":"r = u => Math.exp(u) + Math.exp(-u) + 1","pinned":true,"hidden":false,"mode":"ojs"},{"id":963,"value":"svg`