Gravity from Another Perspective

“OK, we’re looking at that robot next to the black hole and he looks smaller to us because of space compression down there.  I get that.  But when the robot looks back at us do we look bigger?”

We’re walking off a couple of Eddie’s large pizzas.  “Sorry, Mr Feder, it’s not that simple.  Multiple effects are in play but only two are magnifiers.”

“What isn’t?”

“Perspective for one.  That works the same in both directions — the image of an object shrinks in direct proportion to how far away it is.  Relativity has nothing to do with that principle.”

“That makes sense, but we’re talking black holes.  What does relativity do?”

“Several things, but it’s complicated.”

“Of course it is.”

“OK, you know the difference between General and Special Relativity?”

“Yeah, right, we learned that in kindergarten.  C’mon.”

“Well, the short story is that General Relativity effects depend on where you are and Special Relativity effects depend on how fast you’re going.  GR says that the scale of space is compressed near a massive object.  That’s the effect that makes our survey robot appear to shrink as it approaches a black hole.  GR leaves the scale of our space larger than the robot’s.  Robot looks back at us, factors out the effect of perspective, and reports that we appear to have grown.  But there’s the color thing, too.”

“Color thing?”

“Think about two photons, say 700-nanometer red light, emitted by some star on the other side of our black hole.  One photon slides past it.  We detect that one as red light.  The other photon hits our robot’s photosensor down in the gravity well.  What color does the robot see?”

“It’s not red, ’cause otherwise you wouldn’t’ve asked me the question.”

“Check.”

“Robot’s down there where space is compressed…  Does the lightwave get compressed, too?”

“Yup.  It’s called gravitational blue shift.  Like anything else, a photon heading towards a massive object loses gravitational potential energy.  Rocks and such make up for that loss by speeding up and gaining kinetic energy.  Light’s already at the speed limit so to keep the accounts balanced the photon’s own energy increases — its wavelength gets shorter and the color shifts blue-ward.  Depending on where the robot is, that once-red photon could look green or blue or even X-ray-colored.”

“So the robot sees us bigger and blue-ish like.”Robots and perspective and relativity 2“But GR’s not the only player.  Special Relativity’s in there, too.”

“Maybe our robot’s standing still.”

“Can’t, once it gets close enough.  Inside about 1½ diameters there’s no stable orbit around the black hole, and of course inside the event horizon anything not disintegrated will be irresistibly drawn inward at ever-increasing velocity.  Sooner or later, our poor robot is going to be moving at near lightspeed.”

“Which is when Special Relativity gets into the game?”

“Mm-hm.  Suppose we’ve sent in a whole parade of robots and somehow they maintain position in an arc so that they’re all in view of the lead robot.  The leader, we’ll call it RP-73, is deepest in the gravity well and falling just shy of lightspeed.  Gravity’s weaker further out — trailing followers fall slower.  When RP-73 looks back, what will it see?”

“Leaving aside the perspective and GR effects?  I dunno, you tell me.”

“Well, we’ve got another flavor of red-shift/blue-shift.  Speedy RP-73 records a stretched-out version of lightwaves coming from its slower-falling followers, so so it sees their colors shifted towards the red, just the opposite of the GR effect.  Then there’s dimming — the robots in the back are sending out n photons per second but because of the speed difference, their arrival rate at RP-73 is lower.  But the most interesting effect is relativistic aberration.”

“OK, I’ll bite.”

“Start off by having RP-73 look forward.  Going super-fast, it intercepts more oncoming photons than it would standing still.”

“Bet they look blue to it, and really bright.”

“Right on.  In fact, its whole field of view contracts towards its line of flight.  The angular distortion continues all the way around.  Rearward objects appear to swell.”

“So yeah, we’d look bigger.”

“And redder.  If RP-73 is falling fast enough.”

~~ Rich Olcott

  • Thanks to Timothy Heyer for the question that inspired this post.
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Three ways to look at things

A familiar shadow loomed in from the hallway.

“C’mon in, Vinnie, the door’s open.”

“I brought some sandwiches, Sy.”

“Oh, thanks, Vinnie.”

“Don’t mention it.    An’ I got another LIGO issue.”

“Yeah?”

“Ohh, yeah.  Now we got that frame thing settled, how does it apply to what you wrote back when?  I got a copy here…”

The local speed of light (miles per second) in a vacuum is constant.  Where space is compressed, the miles per second don’t change but the miles get smaller.  The light wave slows down relative to the uncompressed laboratory reference frame.

“Ah, I admit I was a bit sloppy there.  Tell you what, let’s pretend we’re piloting a pair of space shuttles following separate navigation beams that are straight because that’s what light rays do.  So long as we each fly a straight line at constant speed we’re both using the same inertial frame, right?”

“Sure.”

“And if a gravity field suddenly bent your beam to one side, you’d think you’re still flying straight but I’d think you’re headed on a new course, right?”

“Yeah, because now we’d have different inertial frames.  I’d think your heading has changed, too.”two-shuttles

“So what does the guy running the beams see?”

“Oh, ground-pounders got their own inertial frame, don’t they?  Uhh… He sees me veer off and you stay steady ’cause the gravity field bent only my beam.”

“Right — my shuttle and the earth-bound observer share the same inertial frame, for a while.”

“A while?”

“Forever if the Earth were flat because I’d be flying straight and level, no threat to the shared frame.  But the Earth’s not flat.  If I want to stay at constant altitude then I’ve got to follow the curve of the surface rather than follow the light beam straight out into space.  As soon as I vector downwards I have a different frame than the guy on the ground because he sees I’m not in straight-line motion.”

“It’s starting to get complicated.”

“No worries, this is as bad as it gets.  Now, let’s get back to square one and we’re flying along and this time the gravity field compresses your space instead of bending it.  What happens?  What do you experience?”

“Uhh… I don’t think I’d feel any difference.  I’m compressed, the air molecules I breath are compressed, everything gets smaller to scale.”

“Yup.  Now what do I see?  Do we still have the same inertial frame?”

“Wow.  Lessee… I’m still on the beam so no change in direction.  Ah!  But if my space is compressed, from your frame my miles look shorter.  If I keep going the same miles per second by my measure, then you’ll see my speed drop off.”

“Good thinking but there’s even more to it.  Einstein showed that space compression and time dilation are two sides of the same phenomenon.  When I look at you from my inertial frame, your miles appear to get shorter AND your seconds appear to get longer.”

“My miles per second slow way down from the double whammy, then?”

“Yup, but only in my frame and that other guy’s down on the ground, not in yours.”

“Wait!  If my space is compressed, what happens to the space around what got compressed?  Doesn’t the compression immediately suck in the rest of the Universe?”

“Einstein’s got that covered, too.  He showed that gravity doesn’t act instantaneously.  Whenever your space gets compressed, the nearby space stretches to compensate (as seen from an independent frame, of course).  The edge of the stretching spreads out at the speed of light.  But the stretch deformation gets less intense as it spreads out because it’s only offsetting a limited local compression.”

“OK, let’s get back to LIGO.  We got a laser beam going back and forth along each of two perpendicular arms, and that famous gravitational wave hits one arm broadside and the other arm cross-wise.  You gonna tell me that’s the same set-up as me and you in the two shuttles?”

“That’s what I’m going to tell you.”

“And the guy on the ground is…”

“The laboratory inertial reference.”

“Eat your sandwich, I gotta think about this.”

(sounds of departing footsteps and closing door)

“Don’t mention it.”

~~ Rich Olcott