Part 2! How did Einstein develop General Relativity? What does it mean for different kinds of masses to be equivalent? How does gravity do what it does? Why is curvature so important in understanding gravity? I discuss these questions and more in today’s Ask a Spaceman!
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EPISODE TRANSCRIPTION (AUTO-GENERATED)
Well, welcome back. It's been a while, hasn't it? It's been way too long since I left you at that wonderful, wonderful cliffhanger. If you haven't listened to the first episode in this series, you probably should because I really, really want you to understand what I'm saying. And not listening to the first part of a multi part series is a serious detriment to that goal.
I mean, it it it's hard to have complete knowledge of time and space, which is this show's goal, without having, you know, complete knowledge of time and space. So if you haven't listened to the first episode, go ahead and do that now. I will be here waiting. Don't worry. I am not going anywhere.
Now I hope you're back. Welcome back. Last time I left you, we learned the state of the world in nineteen o five, '19 o '6, after special relativity had made its big splash, and everyone's saying, oh, now I get it. Now I get how space and time are related. Now I get how mass and energy are related.
Now I get I need to go back through all the physical theories and make sure they're compatible with this new thing that's going on called special relativity. And the one thing that was missing from that, the big one that was missing from that was gravity itself. So people were thinking everyone was thinking, including Einstein, about how to make gravity agree with special relativity. How do you cast the equations of gravity in a form that's compatible with this view of interlocked space and time on non simultaneity of observers about all the juicy good stuff that lives in special relativity? I also gave a mini rant about Einstein and how we perceive Einstein in the present day and how it took years of toil to come up with what we call general relativity or modern theory of gravity.
And I introduced the key, the foundation, the cornerstone, the one absolute, what appears to be fact about our universe that I want you to take away from these episodes, and that is the equivalence principle. Bake that into your brain. Get a tattoo somewhere convenient that you can look at every single day. You can even write it backwards on your forehead so you can see it in the mirror when you get up. This is why I want you to write inertial mass equals gravitational mass.
Inertial mass equals gravitational mass. They're the same thing. This was kind of sort of assumed but unexplained in Newton's gravity, and then Einstein took it one step further. He said, this is just the way the universe works. We're gonna use it to explore the consequences.
And we learned in the last episode that I am taking my sweet time exploring this topic because general relativity is so complex. It's so beastly. It's so mathematical that I really want us to take one chunk at a time, chew on it, digest it, let it soak in, enjoy the moment before moving on to the next. So I'm in absolutely no hurry. I might finish today.
I might not because that's just the kind of journey we're on. We are going to understand general relativity one step at a time. We're all gonna do it together hand in hand, and we're gonna do it at a nice mosey, a mosey pace to understand relativity. Now we got through Einstein's first big thought experiment of how he realized the implication of this equivalence principle of how if you're falling, you're in free fall, you feel zero gravity. And in zero gravity, everything you know about special relativity applies to you.
The interlocking space time, the inertial reference frame, it all it's all there. So when you're studying gravitational problems and you're in free fall, all of a sudden, special relativity pops out. Now the next big leap was this mad scientist lab that I introduced at the very beginning of the last episode where if we stuffed you inside of a rocket and closed all the doors and windows and made it totally soundproof so you didn't hear the rumble of the engines, there were no windows so you couldn't look outside, you couldn't tell if you were still on the ground or if you were in a rocket ship that was accelerating at 9.8 meters per second squared at the same acceleration that the Earth provides through gravity at its surface. You couldn't tell. You would be pinned to the floor.
If you tripped on something, you would fall to the floor. If you drop something off a counter, it would fall the exact same rate you expect. If you poured some milk or some apple juice, it would flow at the exact same rate you're used to. Everything would appear. Your cardiovascular system would be totally okay and fine.
Everything would be the same. Now Einstein didn't use the terminology mad scientist lab. That that's my own, creative addition. Please forgive me, Einstein. But it's the exact same idea of what he was thinking.
And he was specifically thinking about the importance, what really caught him, what were you he really stuck on in a good way was Galileo. Galileo's experiment where he took different balls with different masses and dropped them off a tower and watched to see what happened. And the old way thing is was that if something is heavier, if something feels heavier in your hand, it must fall to the ground faster. And he showed no. A heavy ball and a light ball will drop to the ground at the exact same rate.
This led to the concept of equivalence that's baked into Newtonian mechanics and then raised the level of a principle in Einstein's work. All objects fall at the same rate. Yes. I'm ignoring air resistance. This is thought experiments.
We're making ideal situations. Now imagine recreating that experiment inside a rocket that's blasting away at exactly one g at that 9.8 meters per second squared. From outside the rocket from outside the rocket, you see exactly what's going on. Someone's holding, say, a ball, and they're standing on the floor of the rocket, and the rocket's pushing them around. And then they drop the ball, and it's not you, the external observer.
It's not that the ball falls to the ground. It's more like the ball stays put, and then the rocket travels to meet it. So in your perspective, outside the rocket, of course, everything falls at the same rate because the quote, unquote fall is really caused by the rocket itself itself moving. It's caused by that acceleration of the rocket. It doesn't matter if you drew up a a little grape or a massive weight, a bowling ball.
It doesn't matter what you're dropping. They're just hovering there in space, and then the rocket comes and meets it. And that's determined by how fast the rocket is moving by its rate of acceleration, not by the actual drop of the object itself. But you inside the rocket, you don't have that privileged position where you're peeking in through the windows. You inside the rocket, you just start dropping stuff because you're clumsy and or you wanna be a scientist.
You start dropping stuff, and you notice, like, hey. Everything's falling to the bottom of my rocket exactly the same. Right? The equivalence principle holds. It's right there inside the rocket.
And to you, nothing is different than if you were on the surface of the Earth. If you're Galileo on the Earth, you're dropping ball balls off a tower, they hit the ground at the same rate. If you're Galileo inside of a rocket ship and you're dropping balls off of a tower, inside the rocket ship, they hit the ground, they hit the floor at the same rate. You can't tell the difference. Equivalence.
Equal. You can't tell the difference between an accelerating rocket and the gravity of the Earth. So, of course, Einstein objects must fall at the same rate because of the equivalence principle, and here's the major jump that only Einstein can make. Like, we can think of these log experiments, and we're like, oh, that's pretty cool. That's pretty nifty.
What is that teaching us? Einstein's here to tell us what it taught us. We're used to the statement, gravity causes acceleration. Right? We have the gravity of the Earth, and it's causing acceleration.
Einstein flipped that around. Acceleration causes gravity. I'll say that again because that's kind of a super deep insight that most of us, including myself, probably have never thought before. Acceleration causes gravity is pretty messed up. Gravity causes acceleration, whatever.
Acceleration causes it makes gravity happen. The same way gravity can make acceleration happen, acceleration can make gravity happen. If you're inside that rocket ship, you can't tell the difference between the fact that you're in a rocket ship and you're on the surface of the Earth, you are experiencing gravity exactly the same as if you were on the surface of the Earth. Acceleration is gravity. Gravity is acceleration.
They are linked. They are connected. They are identical. That is Einstein's evolution of the equivalence principle. You can tell no difference.
You can perform no experiment to decide between whether you're sitting on the surface of the Earth or blasting in a rocket. And this is where we get the concept that we view gravity as a fictional force. Remember, if you're fictional force, if you're riding along in a car and someone and the driver turns the the steering wheel really, really hard and you feel pressed up against the side of the car, like your face is all smooshed against the windshield, you know that there's not, like, an invisible hand shoving you into the side. You know it's because you would prefer to go in a straight direction, straight ahead, but the car is turning, and so there's a mismatch of reference frames that are causing the appearance of what you feel as a force. Now there's an actual force there, the the electrostatic force between the the atoms and molecules of your skin and the door and the windshield of the car are pressing up against each other.
That's how you feel the sensation in your nerves and in your brain, but it's caused by the car turning, not by an invisible hand shoving you against the wall. Gravity is a fictional force in the same way. You would prefer to be falling to the center of the Earth, but the ground gets in the way, and the electric compulsion between the atoms of the ground and your feet are what give you the sensation of the force of gravity. So that when astronauts are in a rocket or jet pilots in our in their jets and they're experiencing g forces because of the extreme acceleration, it's identical to the acceleration of gravity, the effects of gravity. And up in a rocket ship or up in a jet, we can easily say, yeah.
You feel pressed against the back of your seat because the jet's accelerating like crazy. And so that's a fictional force. Well, guess what? Acceleration is gravity. The force you feel, your sensation of weight is a fiction.
It's caused by the exact same thing. Gravity is acceleration. Acceleration is gravity. So that's nice. Gravity is acceleration.
And if I happen to be in free fall, if I'm not pressing up against any floors or windshields or dirt or whatever, if I happen to be in free fall, then I reconnect with special relativity. So we've got two clues now. Special relativity is there lurking in the shadows, and gravity is identical to acceleration. Huge world changing insights, but, you know, no theory of gravity. Not quite yet.
That's not like an a mathematical model. But this was in nineteen o seven, by the way, and Einstein was stuck with these two insights without making major progress for another five years. He took another five years to further advance general relativity. He got these ideas right away, and he just chewed through the consequences of these ideas for five years. And it was a fascinating journey where it's like he he felt that these were true statements about the universe, and he was just kinda tossing around, playing with it, massaging it, fleshing it out to see where it might lead him so that he could eventually develop a fully mathematical theory of gravity.
And just from those two statements, you can have all sorts of cool insights about the universe, insights that we usually attach to general relativity, but really these effects are much older. They're based on the fundamental cornerstone principles of general relativity rather than the fully fleshed out theory itself. For example, clocks run slow in strong gravity. Clocks run slow in strong gravity. If you have a moving clock, moving clocks like we talked about in the space time special relativity episode, moving clocks run slow.
Accelerating clocks run slow. Acceleration is gravity. Clocks experiencing strong gravity are exactly the same as clocks experiencing strong acceleration, which is the same thing as clocks moving, which means they run slow. Us here on the surface of the Earth have a slightly stronger gravity than, say, an astronaut or someone 60 miles up, a hundred miles up, a hundred kilometers up, whatever. Our clocks are a little bit slower than theirs.
You put yourself near the edge of a black hole, near the event horizon incredibly strong gravity, your clock will run incredibly slow. This also means that the light coming up from the surface of a massive object will get redshifted a little bit. It will lose energy. Because if you got something generating light, like, I don't know, an electron wiggling back and forth or an atom jiggling up and down. Way down there on the surface, nears a black hole or a neutron star or the sun or the Earth experiencing that strong gravity, it's gonna move a little bit slower than normal.
Just as it's gonna be a little bit lazier because clocks run slow down there. And so the light that you expect it to make will be red shifted. It'll be shifted down and lower. And then, conversely, if light is generated in a low gravitational environment, it'll be like normal speed or what you expect to be normal speed, so it would be tend to be blue shifted. Light going from the floor to the ceiling gets red shifted a little bit.
Light going from the ceiling to the floor gets blue shifted just a little bit. And, yes, we can totally measure it, and I'll talk about that later. Patience, grasshopper. And you get that just from the equivalence principle. Just from the equivalence principle realizing that gravity is acceleration.
You also get the bending of light due to gravity. And you can see this. It's this is why thought experiments, especially Einstein's thought experiments, are so incredibly useful because you can think in the same situation and keep thinking of all these little additions and tweaks, and you gain insights into the nature of reality. In this case, you'll learn that gravity will bend the path of light. Now the analogy I like to use here is, let's say you're getting on and off a subway car.
Let's you're standing on the edge of the track. And let's say this is a a weird subway car that doesn't stop. And there's just one car, and it's just you. Okay. Thought experiment.
Roll with me here. You look down the tunnel, and the subway car comes in, crosses the platform, goes right in front of you, and then leaves. Okay. It's just one car. And let's say there's four doors or there's there's there are doors.
Who who cares about the number of doors on this subway car? Somewhere towards the front, somewhere towards the back. It doesn't matter. Now your goal is you don't want to stay on the subway car. You wanna get to the other side of the track.
So what you're gonna do is when the subway car passes in front of you, you're gonna walk on to that front door. You're gonna walk in a straight line and then exit through a door on the opposite side of the car onto the opposite side of the platform, and then do whatever you're gonna go do. I don't really care, and that's not important because this is the thought experiment. So from your perspective, from your perspective, you're doing the walking. You're just gonna move in a straight line.
You're not gonna turn left. You're not gonna turn right. You're not gonna hop up and down. You're gonna go in a straight line. What do you do?
Well, you take a step. That subway car comes. You see that front door. Boom. You take a step.
Now you're on the subway car. You walk, walk, walk, and then you exit. And you're gonna exit because you want to exit in an exact straight line. You don't wanna deviate left or right on that platform. You're gonna be exiting through the rear door of the subway car.
You entered through the front, left through the rear. What does it look like from the perspective of someone inside the subway car watching you? Well, it looks like you entered through the front door, and then you followed a curving path to get to the back door, and then you left. You thought you were going straight. Someone sitting already sitting in the car thought you were moving with the curved path.
Let's take it back to the rocket ship, Einstein's original thought experiment. Let's say there's a beam of light entering the rocket ship, and it enters through a window at the very, very tip of the rocket. What do we see from our outside perspective? Someone just watching the scenario passively shooting lasers at rocket ships, which would make for an amazing real life science experiment. The laser, the light enters the top window.
Then in the time it takes for the light to cross the width of the rocket, the rocket's gonna move up because that's what rockets do. So by the time the light gets to leave in that microsecond or that two seconds, you know, however long it took that light to get across the body of the rocket, it's gonna exit through a window in the bottom. And to you, that makes total sense. There is nothing weird about this scenario at all. The light was going straight, not turning left, not turning right, not going up, not going down, just going straight.
But the rocket itself moved, and so the light is gonna enter the top window and exit through a bottom window on the opposite side. But what if you were inside that rocket? You would see light come in a top window, and then it would leave through a bottom window. In order to do that, its path had to be curved. It had to curve down to go from the top window to the bottom window.
So goes for acceleration, so goes for gravity. If a beam of light passes by a gravitating object, its path will be bent. Again, this isn't coming from general relativity. This is coming from the equivalence principle, from the precursor, from the prequel trilogy that would eventually lead to the saga of general relativity. Something that tripped up Einstein for a long time was that if you just look at the equivalence principle and how light behaves, it also predicts that the speed of light is not constant, that it can actually slow down in some cases, which, of course, everyone, as soon as he published this and said, hey.
Maybe that speed of light thing, you know, that was made a big deal about it being constant. Maybe it's not so constant anymore. Everyone's like, you have got to be kidding me. You just totally made a revolution insisting that speed of light was constant, and we have measurements to show that that's true. And now you're telling me it's not really constant?
Like, who do you think you are, Einstein? He's like, yes. I'm I'm exactly Einstein. I don't I don't want to get into the physics of that because that turns out to be wrong because his full theories weren't complete yet. So some of this is an example.
Some of the insights that Einstein had did lead him on blind alleys, did lead him to to places that that turned out to not be physically true because he didn't quite yet have the full picture of relativity, and that's one of the reasons he was stuck there for five years. So once again, we're doing great. We've got the equivalence principle. We've got free fall observers noticing that special relativity still holds. We've got acceleration is identical to gravity.
We've got all these cool effects, things that we can measure, things that should pop out of a full theory of relativity, but we don't have that entire theory of relativity, do we? We need one more thought experiment. And this one's a good one. It it's I'm gonna be honest. This one's a little bit tougher to visualize.
It's a little bit tougher to communicate. It's a little bit more mathy. It's maybe half a thought experiment, but it is such a key central feature that clicks into place, perhaps, the most important component of the mathematics of general relativity. So the equivalence principle is the cornerstone. This is the stone that comes next to the cornerstone.
I don't think it has a special name. Maybe it's the keystone, the the one that goes on the top of the arch. I don't know my stone masonry. Someone help me out here. It's an important stone.
Okay? I want you to imagine a merry-go-round, and you're gonna design this merry-go-round, you know, with the horses moving in the circle with the music and stuff. And for some reason, you're gonna put these horsies perfectly nose to tail. So you're you're gonna put them on the outermost edge of this merry-go-round, this carousel, and you're gonna touch the horses perfectly. So there is absolutely zero gaps between any horsies.
This is how you're doing your merry-go-round. It's your merry-go-round, your thought experiment, so this is how you get to design it. Now you're gonna spin that sucker up and not just, like, a little bit like, do do do do do do do, the usual merry-go-round pace. This is gonna be a relativistic merry-go-round where you're gonna spin it close to the speed of light. I mean, we're we're serious here.
We're not we're not fooling around. Now what happens? You can ask just what happens? What would you see if you looked at this merry-go-round from the outside? Well, we know that from special relativity that moving clocks run slow and that moving objects, moving rulers get shortened along the direction of motion.
So if a ruler if you just take a meter stick and blast it by you close to the speed of light and you were to see how long it is as it's blasting might, it would not be a meter long. Might be half a meter, might be a centimeter, depending on how fast it's going. Moving clocks run slow, moving rulers get shortened along their direction of motion, an effect we call length contraction. So as you're staring at this merry-go-round, when the horsies pass directly in front of your vision, they get shorter. They get squashed down, which means gaps open up.
They're not perfectly nose nailed because they're not as long as they used to be when they pass in front of your face, which means you can fit more horses around the edge than you originally thought. Well, so what? What about the diameter? Well, the diameter of the merry-go-round hasn't changed at all. The merry-go-round itself isn't moving.
It's just spinning in place. So its diameter is its diameter. You can measure its diameter. And then when it's rotating, you can fit more horses around the edge than when it was standing still, which means when it's moving, the relationship that we know, know, and love between a diameter of a circle and its circumference doesn't hold true anymore. Right?
In the nonmoving stationary merry-go-round, it's just a circle. The circumference equals pi times the diameter. That's just it. We all learned that in high school. But if it's spinning really fast, the circumference is more than pi times the diameter.
A stationary merry-go-round, I can maybe fit 10 horses. Moving one at 99% of the speed of light, now it's 12 horses, but their diameter is the same. How could this possibly make sense? How can this work? Well, it works because the geometry that describes this scenario, the relationship between diameter and circumference of a rapidly spinning merry-go-round is now dramatic music, non Euclidean.
Non Euclidean. And this is probably the word of the day. Non Euclidean. Feel free to slip that in at an inappropriate or nonsensical time into casual conversation. Like, oh, yeah.
I went to dinner last night, but the but the beef was just a little non Euclidean, and see if it gets you anywhere. Tell me the results. Non Euclidean. So the geometry you learned in high school, triangles, parallel lines, all that is not the whole story. That's the geometry of flat surfaces, which is brilliant, brilliant stuff.
Euclid, kind of a genius. His work has been around for a while, lasted literally through the ages. Euclid, Euclidean, the geometry of the flat. But it's not the whole story. You need to extend the language of geometry to include curved surfaces, like the surface of the Earth.
And I talked about this before when we were talking about the curvature of the universe, how the surface of the Earth is not flat. And I know that's kind of an obvious statement, but it's not flat in a geometrical way. Lines that start out parallel, like lines of longitude at the Equator, eventually they don't stay parallel forever. Parallel forever. They eventually converge.
They meet up. They intersect at the poles. Or if you pick three cities on a globe and draw straight lines between them to make a triangle, the interior angles of that triangle are gonna add up to more than a 80 degrees. Even though every single line is straight, you did everything right, the rule of interior angles at in a triangle adding up to a 80 degrees only applies to flat triangles. Now we got curved triangles.
The geometry of the curved is called non Euclidean. It was developed in the mid eighteen hundreds, and this spinning merry-go-round, besides being dangerous and vomit inducing, is an excellent example of how non Euclidean geometry pops up in some surprising places. In this case, the usual rule of circles, the relationship between diameter and circumference, simply doesn't work anymore. So that's it. That's the final piece of the puzzle.
That is the last thing we need to lock into place to get us general relativity. And these are the puzzle pieces that Einstein had in 1912. So pay attention. We're almost there. We almost have a theory of relativity.
Piece number one. Locally, in any tiny patch, especially observers in free fall, special relativity still holds. So space time is the ultimate stage. It's the ultimate language where gravity is going to play. Number two, acceleration is gravity.
Acceleration generates gravity. So acceleration and gravity are intimately linked. Number three, some accelerations, some forms of acceleration, like a spinning merry-go-round, lead to non Euclidean geometry. So for us, a hundred years later, it should be an easy leap. Gravity is curvature.
There it is. But at the time, in 1912, basically, nobody knew anything about all this non Euclidean geometry stuff. It was worked out in the mid eighteen hundreds, like I said, and then kinda sorta forgotten. It's like a tool or an invention that's just been sitting on the shelf. Everyone thought it was a cool tool or invention, but no one had figured out how to actually use it or apply it.
And and no disrespect to all three I mathematicians in the audience. You do math because it's fun, and and you learn insights into all sorts of cool logical structures. You don't do math because it provides handy tools for physicists and scientists to explain the universe, but thank you anyway. Sometimes physicists end up inventing their own tools like Dirac did when he was confronting the marriage of quantum mechanics and special relativity. That's a previous episode.
And sometimes physicists go digging through the garage to find a mathematical tool that works. Einstein had all the puzzle pieces in 1912. He had all the clues, but he hadn't figured out how to fit them together, how to actually describe all these relationship with mathematics and be able to make predictions. So he worked with a close buddy of his, Marcel Grossman, who was a mathematician, to to flesh out how to mechanically make this all work instead of just being a bunch of cool ideas and thought experiments of how to describe reality. And together, they wrote a paper in 1913 that came this close.
And once again, I'm holding my fingers really close together to getting the final form of g r. They had allowed the derivation. They had allowed the language that we would eventually adopt in the mathematics, but there was a trouble. There was a problem with the theory that they were working with at the time. And the problem was that at the end of the day, whatever advanced, crazy, cockamamie, off the wall idea you have for gravity, it must, at some point, connect with Newton's work.
Right? If I'm throwing a ball in the air, playing baseball or whatever, and tossing a ball to someone, Newton's laws are perfectly 100% adequate and capable of describing that motion, right? So, if I come up with a new theory of gravity that goes bigger and deeper and farther and better, I still have to be able to describe a ball being thrown between two people in a baseball field. I still have to do that. I have to reduce to Newton's laws in simple cases.
In the case of flat space times or weak gravity or slow speeds, You know, in if we're to view Newton's gravity as just a special case of something bigger, then whatever I make bigger has to also work. It has I have to recreate Newton's laws in a low gravity situation. It has to naturally pop out of math my mathematics, and the mathematics that Einstein had in 1912 didn't. They didn't. And it's gonna take him another two years to finally crack it.
Two years. He was this close. Seriously, he was around two or three lines of derivation away before he could finally crack it, and it took him two years to get through that. And it's gonna take us until next time to finally have the final complete picture of general relativity in our hands and explore what it means to test it. How do you test this newfangled concept?
Why should we believe it? So I know it's been a long journey. It's been a really fun journey, and it's not over yet. Thank you to Andrew p, Joyce s, at luft eight, Ben w, Colin e, Christopher f, Mariah a, Brett k, Bryguy the fly guy, at Mark Reap, Kenneth l, Allison k, and at srennick shaw for the wonderful questions about relativity. You'll notice I didn't even do a Patreon pitch because it felt weird to do a Patreon pitch in the middle of a series wedged in between a part one and a part three.
We're two thirds of the way through our trilogy here, folks. Hopefully, it's a trilogy. I I don't plan on making it a four parter, but we'll see where we get next time. But, anyway, patreon.com/pmsutter to help make this show possible. Special thanks to Robert r, Justin g, Kevin o, Justin r, Chris c, and Helgeb.
That's patreon.com/pmsutter. If you haven't checked out one of our Astro Tours, we are going all over the world, and that is no exaggeration. Go to astrotours.co. That's astrotours.co. And also check out spaceradioshow.com for the weekly radio show I do.
Feel free to jump on there. If any questions come up during the podcast and you get really curious about it, then just give me a call, Thursdays 4PM eastern, and we can hash it out on the air. Isn't that fun? Also, you can follow me on Twitter and Facebook and Instagram. I'm at Paul Matt Sutter.
Go to askaspaceman.com. Feel free to give a positive review on iTunes. I appreciate that. And I'll see you next time for more complete knowledge of time and space.