It started with a simple graph: the rotation of the function y=1\/(x2+y2-h2).\u00a0 I didn\u2019t quite know what the metric was for a black hole, but I knew I was close.\u00a0 The familiar well came up, with an asymptotal cylinder of radius h.\u00a0 But something else happened.\u00a0 I forgot it also had a \u201cgoblet\u201d in the middle.<\/p>\n
My more formal mathematical training taught me to throw that goblet away and deal with the \u201creal\u201d and \u201cfinite world outside of a black hole\u2019s event horizon.\u00a0 Then I read about Rahmanujan.\u00a0 Why exactly do we throw these away, anyway?\u00a0 Don\u2019t they tell us something?<\/p>\n
Rahmanujan has gained some importance recently, because the peculiar thing about black holes is they touch infinity at the event horizon.\u00a0 So his mathematics of infinity, of throing away this self-imposed limit on our math, would seem to give us some insights.<\/p>\n
But it goes further than that.\u00a0 Could there be a \u201cnegative world\u201d inside the event horizon?\u00a0 And would it be really all that negative?<\/p>\n
How event horizons differ from planet surfaces<\/strong><\/p>\n Take two planets of equal density.\u00a0 Collide them together to make a new planet that\u2019s still perfectly round.\u00a0 The new radius will be 3root2 or ~1.26 times the old radius.<\/p>\n But what happens to the event horizon of two colliding black holes of the same radius?\u00a0 The new radius will be double<\/em> the old radius.\u00a0 Why is that?\u00a0 That\u2019s because an event horizon is nothing like a planet surface, it is the radius at which light makes a perfect circle around a black hole.\u00a0 So the larger the radius, the less light needs to accelerate.\u00a0 Turns out that once the calculations are done, the radius of a black hole is directly proportional to its mass, and not the cube root like a planet.<\/p>\n This means that the larger black holes get, the less dense they need to be.<\/p>\n Black Hole Sea<\/strong><\/p>\n We keep talking about black holes as some dramatic star crushing into a singularity, like that\u2019s all they can be.\u00a0 But so long as any matter of any form and any volume can pull light into an orbit around itself, it will \u201ccloak\u201d itself in an event horizon.\u00a0 The second thing to remember is black holes are really small.\u00a0 A typical stellar black hole will only be a few kilometers across.\u00a0 Compare that to the Earth which is over ten thousand kilometers across.<\/p>\n So a black hole with the density of Mercury\u2019s orbit will have the density of our atmosphere.<\/p>\n What then, about nebulas?\u00a0 While a nebula is only about a thousandth(?) times as dense as our atmosphere, remember you double the radius and you 2^3 volume, so you only need 1\/8 the density.\u00a0 Black holes get rarefied really quickly. A nebula the size of our solar system could well be a black hole candidate.<\/p>\n Now that we\u2019re predicting supermassive black holes at the center of galaxies, that very well may be how these guys have spawned.<\/p>\n But what of the matter on the inside?\u00a0 We can\u2019t imagine that anything spectacularly instant would happen to the matter.\u00a0 So it means we have this matter, which creates an event horizon, and still has a very nicely independent existence from the world outside of it.\u00a0 But then if it\u2019s inside the event horizon, and is still curving spacetime the same way, it should very well curve (carve?) out this goblet shape that asymptotes to a cylinder at the evnet horizon.<\/p>\n The Universe inside a Black Hole<\/strong><\/p>\n Here\u2019s where it gets really interesting.\u00a0 As an event horizon increases, the density decreases.\u00a0 Turns out that the entire mass of the universe can generate an event horizon that\u2019s about the size of the known universe.\u00a0 And here\u2019s the thing.\u00a0 A goblet shape will create \u201canti-gravity\u201d.\u00a0 Everything inside the goblet will slowly slide towards the event horizon on the perimeter.<\/p>\n And so we just found a novel way to explain accelerating expansion.<\/p>\n It\u2019s counterintuitive to think of us being in a black hole when we associate it with such powerful tidal forces and gravitation.\u00a0 But if you think of an event horizon that\u2019s billions of light years across \u2013 that means light only carves a circle once every few dozen billion years.\u00a0 The acceleration would be negligible to even our most delicate instruments.<\/p>\n And considering light is timeless and dimensionless, it really doesn\u2019t care whether it carves out a circle of a few kilometers of a few billion light years.\u00a0 Its path is just as instant and timeless.\u00a0 And so the curvature should not care either.<\/p>\n The Parallax Paradox \u2013 why a goblet appears flat to us<\/strong><\/p>\n But we still have to account for the fact that, by our best instruments, the universe appears to have flat curvature.\u00a0 But how do we know this?\u00a0 We do it by measuring a large flat \u201cparallax triangle\u201d and seeing if the sides add up to 180 degrees.\u00a0 If it \u2018s less or more, we know it\u2019s curved.<\/p>\n The problem is, plenty of shapes will still give us 180 degrees.\u00a0 A cylinder, for one.\u00a0 Which is what a goblet asymptotes into.\u00a0 So we could very well be drawing this triangle that simply follows a curve path along the goblet that still gives us 180 degrees.<\/p>\n","protected":false},"excerpt":{"rendered":" It started with a simple graph: the rotation of the function y=1\/(x2+y2-h2).\u00a0 I didn\u2019t quite know what the metric was for a black hole, but I knew I was close.\u00a0 The familiar well came up, with an asymptotal cylinder of radius h.\u00a0 But something else happened.\u00a0 I forgot it also had a \u201cgoblet\u201d in the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[335],"tags":[],"_links":{"self":[{"href":"http:\/\/mrbiggs.net\/ron\/index.php?rest_route=\/wp\/v2\/posts\/575"}],"collection":[{"href":"http:\/\/mrbiggs.net\/ron\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/mrbiggs.net\/ron\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/mrbiggs.net\/ron\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/mrbiggs.net\/ron\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=575"}],"version-history":[{"count":1,"href":"http:\/\/mrbiggs.net\/ron\/index.php?rest_route=\/wp\/v2\/posts\/575\/revisions"}],"predecessor-version":[{"id":576,"href":"http:\/\/mrbiggs.net\/ron\/index.php?rest_route=\/wp\/v2\/posts\/575\/revisions\/576"}],"wp:attachment":[{"href":"http:\/\/mrbiggs.net\/ron\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=575"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/mrbiggs.net\/ron\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=575"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/mrbiggs.net\/ron\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=575"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}