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Posts Tagged ‘general relativity’

I was watching Dr. Who the other day and came across a physics mistake so common I thought I’d address it here.  The mistake is this:

Black holes suck you in like a vacuum cleaner!

The setup: in Dr. Who [2.8] “The Impossible Planet”, the good Doctor and Rose meet the crew of a ship who are on “an expedition [to] the mysterious planet Krop Tor, impossibly in orbit around a black hole.” [Wikipedia]  That phrase “impossibly in orbit” made me almost spit out my drink while watching the show.

Black holes have event horizons.  I get it.  Even light cannot escape.  I get that, too.  But why does that mean I cannot orbit a black hole?

OK, time for a little general relativity.  Einstein figured out, between 1905 and 1915, that gravity is “just” a warping of space-time.  Matter causes the space-time around it to curve; the curvature of space-time determines how matter moves (insofar as objects in the absence of gravitational forces follow geodesics).  The formulas that link the distribution of matter to the curvature of space are Einstein’s equations:

einstein_equation

This expression is compact and might seem relatively simple, but it’s not.  Gαβ and Tαβ are components of tensors, which are like vectors, but worse; they’re really 4×4 matrices.  So this equation is not one equation, but 16 different equations, since α and β can take on any of four values each.

What do all those letters stand for?  Gαβ is a component of the Einstein tensor, which tells you about how space-time is curved; the indices α and β can be any of four values in a 4D space-time.  (If you’re mathematically inclined, the Einstein tensor can be related to the Ricci scalar, the Ricci tensor, and the Riemann tensor.)  Tαβ is a component of the stress-energy tensor, which basically describes how matter/momentum/energy/stress/strain is distributed in a region of space-time.  So here’s another way to visualize Einstein’s equations:

einstein_explained

The cause (mass) is on the right; the effect (the curvature of space-time) is on the left.

So what does this have to do with black holes?

One of the first solutions discovered to the Einstein equations is called the Schwarzschild solution, which applies to a spherically symmetric gravitational source.  The solution gives you a “metric” (essentially, a geometry) that is almost the same as “flat” space-time, except for a pesky (1–2GM/c2r) term.  But that pesky term has a strange implication: when that term equals zero, the solution “blows up” (i.e. becomes infinite).  Space becomes so curved that you essentially have a hole in the fabric of space-time itself.

When does this happen?  It happens when R = 2GM/c2, as one line of algebra will show.  This is called the Schwarzschild radius.  The Einstein equations predict that something weird and horrifying happens when a mass is squeezed down to the size of its Schwarzschild radius.  Current understanding is that the mass would then keep going, and squeeze itself into a point of zero radius.  Literally, zero.  (I did say it was weird and horrifying).

Incidentally, the Schwarzschild radius is exactly the radius you’d get if you set the escape speed for an object equal to the speed of light.  So this means that not even light can escape this super-squeezed object.

And here’s where various misconceptions start to creep in.

Another name for the Schwarzschild radius is the event horizon.  It’s a boundary of no return:  if you cross it, you can never go back.  But that’s all it is: a boundary.  There is not necessarily anything physical at the event horizon.  You might never know that you had crossed it.  Remember, all the mass is at the center.

Here’s how I “picture” a black hole:

black hole

Now, if I am outside the event horizon, what would I see?  Well, nothing from inside the event horizon could reach me (hence the term “black”) but I might see Hawking radiation.  I would certainly see gravitational lensing: the bending of distant light around a black hole.  Here’s a cool picture of gravitational lensing in action (artists conception only!) from Wikipedia:

225px-Black_hole_lensing_web

Let’s say the Sun were a black hole.  Its event horizon would be around 3 km.  As long as we never got closer than 3km, we could do what we like.  We could fly in, fly out, orbit the black hole as we please.

Would the black hole “suck us in”?  Sure, in the same way that the Sun sucks us in already.  There is a strong pull of the Sun on the Earth.  And there would be a strong pull on our hypothetical spaceship.  But change the Sun to a black hole, and the pull would not get any stronger.  That is the key point that most people miss: black hole gravity is not somehow “stronger” than ordinary gravity.  There is just gravity; that’s it.  Change the Sun to a black hole, and the Earth would continue in its orbit, and nothing would be any different.  Except for, maybe, the lack of light.

Why was the planet Krop Tor’s orbit impossible?  Astronomical black holes (created by stellar collapse) have a lot of mass; when there’s a lot of mass hanging around, things tend to orbit them.  That’s what you’d expect.  It would only be impossible if somehow the orbit crossed the event horizon multiple times during its trajectory.  But of course, the show didn’t mention this.

I want to end my rant on GR with a suggestion: that there are two kinds of sci-fi: science fiction, and “sciency” fiction.  The first kind tries to get the science right, and makes an effort to be possible (if not plausible).  The second kind throws sciency words around in an effort to appeal to a certain demographic.  Basically, “sciency” fiction is fantasy, set in outer space.  When seen in this light, Dr. Who has more in common with Lord of the Rings than it does with 2001.

Don’t get me wrong: I love Lord of the Rings, and I love Dr. Who.  Just don’t call it science fiction.

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Poor Einstein.  Is there anyone else who is misquoted more often?  Is there anyone else to whom more nonsense is attributed?

I have no desire to rehash things that Einstein said about “God”.  Einstein was by all accounts an atheist, an agnostic, or a pantheist—depending upon your definitions—and various religious apologists have been trying to co-opt the man for years by misquoting him.  Others have already discussed this at length.

My goal today is to tackle that old chestnut, “Imagination is more important than knowledge,” as seen on T-shirts, bumper stickers, and even on the packaging of the Albert Einstein action figure.  Did Einstein really say this, and if so, what did he mean?

Here’s the quote in context:

“At times I feel certain I am right while not knowing the reason.  When the [solar] eclipse of 1919 confirmed my intuition, I was not in the least surprised.  In fact I would have been astonished had it turned out otherwise.  Imagination is more important than knowledge.  For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.”  [From A. Einstein, Cosmic Religion: With Other Opinions and Aphorisms, p. 97 (1931).]

So Einstein did say this.  However, I maintain that the full quote in context has a different feel to it than the quote in isolation.

When I see “Imagination is more important than knowledge” on a bumper sticker, I think this: “Flights of fancy and imagination are more important than learning stuff.  So why should I study?  Einstein didn’t study.  He just sat around and daydreamed and came up with the most remarkable breakthroughs about the workings of our universe.  Imagination is more important than learning all the proofs and figures ranged in columns before me.  So I am going to follow good ol’ Einstein and daydream about being Batman.”

The New Age meaning of the quote is this: “I’d rather daydream than study.”  It’s Walt Whitman’s “learn’d astronomer” nonsense all over again.

In context, it’s clear that Einstein was talking about doing science.  Imagination is more important in making scientific breakthroughs than knowledge, but that doesn’t mean that knowledge is not important.  Einstein worked very, very hard to learn an awful lot of physics.  By all accounts, it took him almost 10 years to flesh out general relativity, during which time he had to acquire a lot of mathematical knowledge about Riemannian geometry and tensor analysis.  The “intuition” that Einstein developed during this time frame is what allowed him to be so confident of the results of Eddington’s expedition.  What Einstein calls “intuition” is just knowledge that has become so ingrained that you are no longer cognizant of it.

Einstein may have been more famous than most of his contemporaries, and it was probably due to his superior imagination.  But take Einstein’s imagination today and give it to a twenty-five year old high school dropout, and he’d be lost in obscurity, stocking shelves at Wal-Mart.  Imagination is more important than knowledge.  But only slightly more.

[Note: my book Why Is There Anything? is now available for download on the Kindle!]

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