Hi, in Stephen Hawking's entropy of blackhole equation, where s is entropy and r is the surface area of the blackhole, what is the 4 for? I thought this would be the right [lace to ask!
http://arxiv.org/abs/gr-qc/0010086 Nice pdf, that? Lotta reading. Well the simple answer is the 4 is for 1/4. Yeah, I know, is it really that simple? Essentially, The entropy of a Black Hole is directly proportional to the surface area where "a" is "area" and the radius of a black hole is twice its mass. Draw a circle- draw each radius outward- and you have divided the circle into four. a/4 gives you each quadrant.
The 4 comes from the way in which the surface gravity of a black hole is defined, which appears in the same expression. When doing spheres there's a 4pi somewhere and it comes from that. I'll explain more when I have time.
It follows from the laws of black hole thermodynamics, which attempt to cast various black hole parameters in the same form as thermodynamical ones. The first law shows that regardless of where on the surface you looks, including for spinning black holes which aren't completely spherically symmetric but their event horizon is, there's a quantity you can equate to 'surface gravity' which is the same everywhere. This is \(\kappa\). It's, if I remember correctly, the acceleration a ship at infinity would have to apply if you lowered an infinitely long but massless (and unbreakable) rope down to the event horizon and tried to hold a 1kg mass there. It has to be at infinity for various technical reasons to do with different people seeing different things elsewhere. Anyway. In all of this you also get an expression for the radius of the event horizon, which is spherical for all black holes in 3+1 dimensions, in terms of mass, spin and charge. If you then ask "What happens to the event horizon area if I vary the mass, spin and charge by small amounts?" then you can work out an expression for the 2nd law of black hole thermodynamics. When you do the actual algebra and rewrite a few things in terms of \(\kappa\) you find that there's a term which looks like something times by the event horizon area (which is just \(4\pi\) times the square of the radius as it's a sphere). If it's to look like T dS, as from thermodynamics, then it means you factorise the term such that \(dS = \frac{1}{4}dA\), so \(S = \frac{A}{4}\). If you really want to see the algebra I have all of this done step by step in a set of 4th year lecture notes I transcribed when I was sitting a course in black holes about 6 years ago (god, 6 years.....) which I can upload.
Wow that is really epic. I appreciate the response. I don't think i'm really capable of appreciating the algebra but i would look if you did feel like putting it on the site. I mean I was obviously having trouble with s=a/4 on it's own. so the 4 has nothing to with 4 spatial dimensions or anything. i will read your link soon alphanumeric, after sleep and coffee! and maybe I will post more if I can make sense of anything! and thanks too neverfly!
For contributing how not to go about thingsPlease Register or Log in to view the hidden image! Well... In my defense, that was how it was explained when I had asked that about a year and a half ago (Not here). I actually did google around to see if I could find anything before posting and didn't find much that dealt with the mathematical aspect- but that arXiv paper deals with when it applies. Part of the trouble of not knowing the math... I figured if it was wrong, a correction would arise in short order. Thanks, Alpha Numeric. Shoot if Alpha Numeric wants to upload lecture notes- More power to him!
Not in that direct manner, ie if you were in D dimensions you wouldn't have a 1/D term in there. However, the dimensionality does indirectly matter. The boundary of a 3d region is 2d and the area of the 2d area of the event horizon. If you make a black hole in say 5 dimensional space you can get all kinds of weird and wonderful configurations because the heavy restrictions of 4 dimensional space-time don't apply. In 4d a black hole is entirely defined by it's mass, charge and spin. In 5d you can make black holes whose event horizons look like Saturn's rings and then put a spherical (the 5d notion of spherical) in the middle to make a system which looks like Saturn! Obviously these will then have very complicated boundaries, which are their version of event horizons, and it'll lead to a different specific expression for black hole thermodynamics. The calculations become vastly more complicated than in 4d, which are bad enough.
well that's very interesting guys. the idea of a ring shaped blackhole, forgive my ignorance, but is a kerr singularity in a blackole meant to be torus shaped? does that mean the kerr singularity might be existing in 5 spatial dimensions? also, if 2pi r is a circle and 4 pi r is a sphere, is 3 pi r some kind of cresent shape, or maybe a torus?
The Kerr singularity is a ring, since a point cannot have angular momentum. The event horizon is still spherical though. The fact a Kerr singularity is a ring in 4d doesn't immediately say anything about 5d. The structure of the singularities follow a number of constraints due to the nature of the Einstein field equations. 4d has a lot of constraints which 5d doesn't. You've gotten a bit confused there. A circle has boundary length \(2 \pi r\) and disk area \(\pi r^{2}\). A sphere has boundary area \(4 \pi r^{2}\) and ball volume \(\frac{4}{3}\pi r^{3}\). More generally if an N-sphere (a circle is a 1-sphere and a sphere is a 2-sphere) has volume (or rather it's N dimensional generalisation) \(V(r)\) then it's boundary will have size \(\frac{dV}{dr}\), ie \(2\pi r = \frac{d}{dr}(\pi r^{2})\) and \(4 \pi r^{2} = \frac{d}{dr}(\frac{4}{3}\pi r^{3})\). These all relate to the size of spheres and balls. An (N-1)-sphere is the boundary of an N-ball. ie a 2-sphere is the surface of a 3-ball and a 1-sphere (circle) is the perimeter of a 2-ball (disk). A torus is defined by the product, in the toplogical sense, of two or more spheres. For example, you make the 2 dimensional toroidal surface by getting a 1-sphere, aka circle, and sweeping it around in a circle in a perpendicular direction, giving the donut shape. The same principle is how you make a 3-torus, you take a 2-torus and sweep it out in a circle in a perpendicular direction (you need to have 4d space to see it properly). Generalisations of tori come about when you don't use circles to sweep out circles but say spheres to sweep out spheres.
i appreciate the feedback AlphaNumeric, this is a quality website. i'm just wondering in the 5d model of the blackhole you mentioned is the idea of hyperspheres being proposed as a theoretical construct, a kind of game from the maths and physics, like godel's time machine solution from einstein's general theory of relativity, or as an actual cosmological phenomenon, like a prediction of something possibly somewhere in the universe?
