Some time back in the’90’s I wrote a document explaining some things about black holes. To my amazement, people still read it, and they occasionally send me questions as a result. I’m happy to answer these when I can, and as long as I’m answering them anyway, I might as well post them here.

The latest is from Chris Warring:

My friend and I are having a debate over the question “If the Sun turned into a black hole, what would happen to the Earth’s orbit?”

I quoted from your article http://cosmology.berkeley.edu/Education/BHfaq.html “What if the Sun *did* become a black hole for some reason? The Earth and the other planets would not get sucked into the black hole; they would keep on orbiting in exactly the same paths they follow right now….a black hole’s gravity is no stronger than that of any other object of the same mass.”

My friend argued that since astroids impact the Sun then they would also impact the black hole. This would eventually increase the mass, increase the gravitational pull on the Earth, and place the Earth on a decaying orbit.

I have since read a little on Hawking Radiation, and that black holes evaporate. I now wonder if the black hole that was our Sun would evaporate, losing gravitational effects on the Earth, and the Earth would end up drifting away from where our Sun use to be.

Here’s my answer:

First, let me say that all of the effects you mention are *very *small. They would alter the Earth’s orbit a little bit over very long times. When I wrote what I did about the Earth’s orbit, I wasn’t considering such tiny effects. But they’re fun to think about, so here goes.

It is true that, if the mass of the Sun (or black hole, whichever is at the center of the Solar System) goes up, then the Earth’s orbit will be affected. Specifically, it would move to a smaller orbit. And of course the reverse is true if the mass goes down.

First, let’s talk about what’s happening right now, and then consider what happens if the Sun turned into a black hole. Right now, things do crash into the Sun from time to time, increasing the mass of the Sun. On the other hand, there’s constant evaporation from the Sun’s atmosphere (as well as energy escaping in the form of sunlight, which translates into a mass loss via *E* = *mc*^{2}). I’m pretty sure that the net effect right

now is that the Sun is gradually losing mass. Taken in isolation, this mass change would cause the Earth to drift gradually into a larger orbit.

That phrase “Taken in isolation” is important. There are other things that affect Earth’s orbit much more than this tiny mass loss rate. The main one is gravitational tugs from other planets, especially Jupiter. I

guess it must be true that the gradual mass loss of the Sun gradually makes all of the planets drift further out, although the details might be complicated.

There’s also the fact that the Earth is being bombarded by meteors. Those presumably slow the Earth down in its orbit. Taken in isolation, that effect would make the Earth spiral in towards the Sun.

I’ve never tried to work out the size of any of these effects. A lot is known about the effects of other planets’ gravitation on our orbit (the buzzword for this being Milankovich cycles). The other effects are much smaller.

Now, what would happen if the Sun became a black hole? Things like meteors would still get absorbed from time to time, but much less often than they do now. That may go against intuition, because we think of black holes as really good at sucking things in, but in fact the black hole has the same gravitational pull as the Sun on objects far away, and it’s a much smaller target, so fewer things actually hit it. So the rate

of mass increase due to stuff falling in will be less than it is now. On the other hand, stuff won’t be evaporating nearly as fast as it does now. (There would be Hawking radiation, but that’s incredibly small, much less than the rate at which atoms are boiling off the Sun now.) So the net effect would certainly be that the black hole would gradually go up in mass, whereas the Sun gradually goes down. The net result would be that the Earth would gradually get closer to the black hole.

But again, the key word is “gradually”: these are really really tiny effects. I’d bet that they’d be too small to have any noticeable effect even over the age of the Universe.

This would not be the best outcome, or could it?

Embarrassing. I came up with a question when I was a kid that I used to measure people’s intelligence: “What would happen to the Earth if the sun instantly turned into a stable black hole of equal mass?”

Idiots: “earth would get sucked in”

Smart people: “earth would continue in same orbit”

Geniuses: “earth would drift away immediately”

The reason being that for all intents and purposes into the practical future, the black hole would have the same mass as our sun, but since it would have higher gravitational density (greater concentration of gravitational energy) it pushes into the higher dimensional fabric of spacetime more sharply. The singularity (center of gravity when talking about things like black holes) would actually be pushed farther away. Along 3-dimensional geodesics it would effectively be farther away from the Earth than the sun is. Significantly farther away.

This is something I would only expect someone who had a mastery of general relativity to understand. And the reason why so few get it correct.

I can’t blame you, Ted, for not getting the answer right since you obviously have very little understanding of the more advanced modern physics. But the fact that I understood enough theoretical physics to correctly understand this point when I was five and you still don’t is disheartening.

In case anyone’s wondering, I don’t have much of a comments policy, because I don’t get very many comments. I delete obvious spam, but other comments are generally approved.

I mention this now just in case anyone’s wondering about the previous comment. The physics in this comment is, for the record, incorrect, as nearly a century’s worth of general relativity textbooks will confirm.

I hesitated about posting the comment, not so much because of the incorrect physics as because of the snarky tone (combined with the incorrect physics). I probably would have rejected it if the snark had been directed at someone else, but since it’s directed at me, I don’t much care.

Please explain how it’s incorrect?

I assure anyone reading that it’s 100% correct, though it is a more subtle aspect of the geometric implications of GR.

Yes my tone was rude. I was in a “snarky” mood. And misunderstanding of this concept is one of my pet peeves.

But that aside, please explain why you think it’s incorrect and I’ll try to explain.

There is a very basic theorem which states that a spherically symmetric mass has the same effect as a point mass of the same mass at the center of the sphere. So Ted’s right!

No I never disputed that, Phillip. This is about a different consequence of being a black hole. The mass can be treated as the same, but it is farther away. The center of mass of the black hole is farther away than the center of mass of the star.

It’s due to the change of the geometry of spacetime.

One of the properties of space time that is affected by energy-momentum is space length and curvature. The gravitation still propogates along the shortest path, but this shortest path is not as short as the distance between the centers of gravities of the earth and the sun were before the star’s collapse into a black hole.

What you get is a spacelike geodesic whose norm becomes even greater. Whereas if it were the converse (the manifold’s effect on the energy-momentum) then you’d have a timelike geodesic whose norm became lesser.

The simple way to visualize it is the old rubber plane analogy where masses are represented by marbly things causing depressions. The sun collapsing into a blackhole represents a really dense marble that stretches the rubber. The stretch represents the increase in distance. Since the Earth represents the optima of an evolution at that orbit, less gravity means it will no longer have a stable orbit and will spiral away in increasingly eccentric elliptic curves.

You can kind of (but not really) think of the gravitational energy density of a body being a metric by which you can tell if its exceeded the threshold where it will stretch rather than merely warp space. And that threshold is also where outward pressure can’t stop a star from collapsing.

Another way to look at it (and this is an oversimplification) is that a black hole is a push into a higher spatial dimension. That’s why you can approach a black hole from ALL three degrees of freedom and get sucked in. People have the wrong conception of this too. Usually because of the limit of ability to draw the acretion disc properly.

Also, regarding Ted’s comment:

“The physics in this comment is, for the record, incorrect, as nearly a century’s worth of general relativity textbooks will confirm.”

Unless the field equations for GR have changed significantly, a century’s worth of textbooks would actually yield exactly the result I’m talking about.

You could, with some fairly contrived use of language, argue that the Sun has become farther away from the Earth, but I don’t think there’s any reasonable sense you could say that the Earth has “drifted away.” The Earth’s orbit is precisely the same geodesic, with the same circumference, length of year, etc. Every measurement that can be made from outside of the (original) surface of the Sun will be exactly the same.

The sentence “it will no longer have a stable orbit and will spiral away in increasingly eccentric elliptic curves” is unambiguously false for precisely the reason Phillip Helbig mentions. Look up Birkhoff’s Theorem for details.

I don’t think you’re understanding what I’m saying.

I fully agree that a spherically symmetric mass can be treated as a point mass (of the same mass) for gravitational reasons.

But the nature of a black hole is that it has become farther away. Spacetime is stretched in this regard. The effective distance between the earth and the center of mass is greater.

Space is no longer the same in the neighborhood of the sun’s mass. It can no longer be, gravitationaly speaking, be treated as a black box mass with no outside effects. The lines along which gravitational force propagates are now longer.

Space is not a three dimensional manifold. It has more than three dimensions and some of the higher dimension are ones in which the energy-momentum has its manifold stretching effect.

Look at it this way: if you (ignoring being ionized etc.) could just fall from the surface of the sun to its center, you would travel a distance approximately equal to the radius of the sun.

But if you could fall into a black hole, the distance would be many times as much. The singularity is a finite distance away from that same starting point, but it is many times as much as the radius of the sun. This is what is meant by the “collapse”

This is of course ignoring the fact that from an outsiders reference frame, you can never reach the singularity, but for you (ignoring spaghettification, etc.) you absolutely can. It’s just a long long fall.

This fall is into a higher dimension. And that’s why you’re able to fall into a black hole from any direction.

This has nothing to do with what happens to the Earth’s orbit. The Earth’s orbit is a geodesic in the spacetime lying exterior to the (original) Sun’s volume, which is at all times Schwarzschild spacetime (via Birkhoff’s theorem). Unless you can show that the spacetime in which the Earth moves is not Schwarzschild, or you can show that there is a solution to the Schwarzschild-space geodesic equation corresponding to a body “spiral[ing] away in increasingly eccentric elliptic curves,” your statement is mistaken.

(If you do come up with either of these things, don’t waste your time posting it here. Publish it somewhere, as it’ll be the most important result in general relativity in 70 years.)

We’ll just have to agree to disagree at this point.

All I’m saying is that the mass will be farther away from the earth due to stretching. Gravity is now propagated over a greater distance and as such pulls less.

It’s a pretty fundamental consequence of GR.

I’ve been working on a GR simulator for awhile now. This would be one of the things that could be verified on it. And of course I’ll upload a video of this exact scenario someday.

In the meantime, the field equations do support what I’m saying though.

Do I understand correctly that you retract the ““spiral[ing] away in increasingly eccentric elliptic curves” language? If so, then we have a semantic difference that I don’t much care about (at least at the moment). If not, then show me an equation for the geodesic describing such spiraling. I promise, all of the geodesics in Schwarzschild spacetime are well-understood, and none of them can be described in this way.

There’s no need for a simulator in this situation. This is one of the very few situations in general relativity in which the equations can be solved exactly.

No I do not retract that.

Do you agree that if the sun suddenly decreased in mass, the earth would spiral away?

“We’ll just have to agree to disagree at this point.”In another context, Ted said that one can’t argue with a theorem. One cannot agree to disagree if one point of view contradicts a theorem. (To be sure, if you could show that an accepted proof of a theorem were false, that would change the game, but in the case of Birkhoff’s theorem that looks unlikely.)

I think I know what you are saying, but it won’t affect the orbit of the Earth. Think of the rubber sheet. Yes, a black hole will push it farther down than the Sun will and, due to the stretching, the centre will be farther away. However, at the Earth’s orbit (or anywhere outside the original surface of the Sun), the slope, which corresponds to the strength of gravity, stays the same. You seem to think that gravity would weaken at Earth’s orbit because, in some sense, the centre of mass is now farther away. However, this misconception is based on using the inverse square law for Newtonian space within the context of curved space. Try it with a rubber sheet. Or, roll marbles at one end of a trampoline and see if you can determine, without looking, if a person at the other end is sitting on the trampoline or standing on one toe.

What about binary systems where one member is a black hole formed from a collapsed star? The other member is still there. Why didn’t it spiral away?

Phillip’s got it exactly right. Just to recapitulate the main points:

Birkhoff’s theorem says that the only spherically symmetric vacuum solution to the Einstein equation is the Schwarzschild solution. That means that, as long as the space surrounding the Sun is empty (no matter streaming outwards, for instance), the spacetime around the Sun remains Schwarzschild spacetime before, during, and after the collapse to a black hole. Moreover, the mass of the system (i.e., the mass that shows up in the Schwarzschild solution) doesn’t change — because a solution with time-varying mass isn’t a Schwarzschild solution.

So the spacetime in the entire region exterior to the Sun remains unchanged during the transition. That means the Earth’s orbit remains unchanged, since it’s completely determined by the spacetime geometry through which it moves.

Every step in that chain of reasoning has the status of a rigorously-proved mathematical theorem. There’s just no wiggle room.

To answer David’s last question: if the Sun suddenly decreased in mass, the Earth’s orbit would change. The change would occur while the mass change was happening — once the mass change was over, the orbit would be a new elliptical orbit (if the mass change wasn’t too drastic) or a new hyperbolic orbit (if it was). The situation is virtually the same as in Newtonian physics, with only minuscule relativistic corrections.

That’s not relevant to the present discussion, though, because the mass of a system doesn’t change during a collapse to a black hole. (How do I know this? Birkhoff’s theorem yet again! Unless matter flows out through the previously-empty space outside the Sun, the space outside the Sun remains Schwarzschild with fixed mass.)