Black Hole with Mass m vs. Plain Old Star with Mass m

Why is it that a black hole gets to do all kinds of special things (warp spacetime, create daughter universes, skip school) that a star of similar mass doesn’t get to do? If gravity is responsible for all a BH’s superpowers, and gravity is directly dependent upon mass, then why doesn’t a regular star with the same mass as a black hole get all the same superpowers? Why is density (apparently) so important? I thought mass was mass and who cares how it’s spread out.

Simple answer: gravity is a function of mass, but also an inverse function of distance. With a giant star and a huge radius equal to a significant chunk of our solar system, there’s not enough mass close together to really get huge amounts of gravitation at any particular point. Put all that mass within a radius of a few kilometers or whatever the Schwarzchild radius of a typical black hole is - then you’ve got all that mass at a much smaller distance, thus the gravity is stronger. I’m not sure about superpowers, but that definitely contributes to the freaky effects of black holes like distorting time.

Since only mass inside the blackhole contributes to its gravitation, you only have a blackhole if the Swartzchild radius is larger then the radius of the object. Blackholes with star-like masses only have radii of about the size of Earth, while stellar densities mean that the size of such objects will be around the radius of the sun.

The exterior of a star, gravitationally, looks pretty much identical to a black hole and indeed in general relativity the solutions that describe black holes are also used to describe the exterior regions of a stars. Obviously though once you get in to the interior region of the star you have to use a different solution. In other words normal stars and black holes warp spacetime in exactly the same way, up to the surface of the star.

In GR the concept of mass isn’t the defining feature of gravity as it is the distribution of stress-energy that decides what form a gravitational field should take. In fact in GR there is no general way of defining the mass of objects or the total mass contained in a spacetime. In certain types of spacetimes though, e.g. asympotically flat spacetimes, there are several definitions of mass.

I saw what you did there.

If an object’s Schwartzchild radius is bigger than itself, all its mass will collapse into a singularity — a point-like anomaly in spacetime which has infinite density around which gravity creates an “invisible” spherical barrier called an event horizon wherein the curvature of spacetime is so warped, all geodesics (even for the speed of light) curve back toward the singularity.

No singularity, none of the cool stuff like smoking behind the bleachers.

As has already been pointed out, gravity depends on both mass and distance, and you can theoretically have black holes of any mass. What makes plain old masses different from black hole masses is that before you get close enough to the plain old mass for gravity to get so strong all the weirdness happens, you hit the plain old mass and gravity stops increasing. Dig a hole in the Earth for instance and gravity is lower at the bottom. A black hole is very tiny compared to its mass. It’s tinier than a neutron star, and in a neutron star the entire star is squashed to the density of the nucleus of an atom. This means you can get closer and closer and closer and gravity just keeps on increasing and eventually weirdness happens.

Btw I’ve been trying to think of a short easy answer to your question. Black holes are inherently general relativistic objects, there’s no real analogue in Newtonian gravity (yes so-called ‘dark stars’ may seem very much like black holes, but the similarity is mostly superficial), so comparisons with Newtonian gravity will not be informative.

A black hole is defined by the existence of an absolute horizon aka the event horizon. The existence of an absolute horizon very trivially depends on the properties of the whole spacetime, because if there is just one worldline leading from the region of spacetime bounded by the horizon to the region of spacetime outside of the horizon, then it is not an absolute horizon. Therefore it is not surprising at all that whether there is an event horizon is sensitive to the exact distribution of matter as the distribution of matter defines the spacetime.

Unsurprisingly, I lost you.

Every moment in space time has an event horizon (I’m thinking of the outer lines of a light cone). How can an event lead to a region of spacetime outside of the horizon? Is this a special super-duperpower of a black hole?

This is very true and I should’ve used ‘scare quotes’ around horizon, it’s also exactly why it’s so hard to define a black hole mathematically. The answer is that some regions are causally-connected to future null infinity and some aren’t. The region that is lies outside the event horizon of the black hole and the region that isn’t lies within the even horizon. Of course you then face the problem that not all spacetimes have a future null infinity.

<nods, takes another hit>

Actually, I hope you guys keep talking about this. Gotta look up “future null infinity.”

Think of it this way: it’s often useful to perform a two-point compactification of the real numbers by adding the two points +∞ and - ∞. This allows you study the real numbers as a subspace of a compact topological space.

In general relativity it is often useful to study a (non-compact) spacetime as a submanifold of a compact manifold by conformally (i.e. in a manner that preserves much of it’s geometric structure) embedding it in to a compact spacetime. The reason for this is that if a spacetime has asympotic symmetries, such as an asympotically flat space time, then the asympotic symmetires will be realised at the boundaries of the embedding.

If that’s a bit too mathematical then, null future infinity is where all the rays of light that escape to infinity end up in the infinite future (though it’s slightly more nuanced than that)

The concept of “future null infinity” is probably best explained using Penrose diagrams. A Penrose diagram is a way of schematically representing all of spacetime, past present and future, in a finite diagram. The first image from that page shows a Penrose diagram for boring old infinite flat space. A photon (or other massless particle) anywhere in spacetime follows a 45 degree slanted line, and hence ends up hitting somewhere on one of the two upper sides of the square: Those sides are “future null infinity”. Any massive particle follows a path that, on that diagram, curves, so that it eventually hits the top vertex of the square: That’s “future timelike infinity”. The past infinities are analogously defined: They’re where things come from, instead of where things end up. And nothing comes from or goes to spacelike infinity, the two vertices on the sides of the diagram.

OK, that’s the flat space one. For a Universe containing a black hole, though, there are some points in spacetime from which light will not ever reach future null infinity, but will instead inevitably hit a singularity. Any such point is by definition inside the event horizon, and any point where you can reach future null infinity is outside the event horizon. But you have to consider the entire spacetime to make this determination; locally, there’s not actually anything special about the event horizon.

My understanding is that if the earth were a black hole (imagine it being the size of a basketball) and you were 4000 miles from its center, it would be exactly like being on earth. Of course, you have to accelerating at 1 G to maintain that distance… Perhaps this is slightly off in general relativity, but only slightly.

Slightly off-topic question: How hot is a black hole? It’s essentially a star, so I would expect it to be very hot. But if it absorbs light, wouldn’t it also absorb heat?

It’s actually dead on in general relativity (as long as you make the necessary approximations). The Schwarzschild solution is the unique spherically symmetric vacuum solution in general relativity, so for any spherically symmetric object considered in isolation, it’s gravitational field in the vacuum will be the same as some Schwarzschild black hole’s.

Amazing stuff. (Plus I enjoy graphical systems in general, and took a side detour to Prof. Croation guy at Georgia, as skipped-scopped by Wiki.).

Will cogitate and enjoy. Perhaps then excogitate here.

Black hole thermodynamics is a big topic itself, it’s temperature is related to a quantity sometimes called it’s ‘surface gravity’, which basically how quickly a black hole accelerates objects at it’s event horizon, as viewed by an observer at infinity.

As a rule, the more massive the black hole, the lower it’s temperature. For a black hole that is a stellar remnant, it’s temperature will always be a tiny fraction of a Kelvin above absolute zero. For a tiny black hole though, it’s temperature can become very large indeed.

Interesting. Thank you!

Well, at least until that inconceivably distant time when the trickle of radiation has finally managed to shrink it to a hotter size.