Physics: Relativistic Momentum

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Einstein's special theory of relativity tells you about how space and time have to be modified when systems or particles are moving close to the speed of light. You can think of this, if you like, as the generalization of classical ideas of kinematics, which describe space and time. How about dynamics? How about when we want to talk about forces and momentum and energy? And can we come up with some relativistic generalization of these ideas? The answer is yes. Einstein was able to generalize these ideas, but you can't just use the old non-relativistic formulas. Let's think about momentum. The non-relativistic formula says momentum of an object is mv. Let me call it m[0]v - mass of the object times its velocity. Now, one of the cornerstones of classical physics is conservation of momentum. It's a fantastic idea in physics, which is very useful for understanding how things work.
Conservation of momentum does not mean that momentum is the same for all observers. And look at the formula. Velocity is a relative concept. In different reference frames, an object will have different velocities so it will have different momentum. That's not what conservation of momentum says. Conservation of momentum says if you have an interaction between two particles and they don't have any external forces acting on them, they collide and recoil. Conservation of momentum says the total momentum of the system before and the total momentum of the system at any time after will be equal. That's conservation of momentum. And you would think that if momentum was conserved in one reference frame, it really better be conserved in any reference frame. And if you use this formula and you look at some collision where momentum is conserved. And then you shift into another reference frame. So you've got to use the Lorentz equations, which tell you about the new positions, and then you can use the new velocities. You can use velocity addition formula, if you want.
And so you work out the velocities of all of the particles before and after in the new coordinate system. You will discover that momentum is not conserved. It's quite a tedious exercise, but if you use this non-relativistic formula, it just doesn't work in the sense that momentum will not be conserved when viewed in different reference frames.
So we'd like to fix this up, and Einstein was able to fix it up in a very simple way. Here's the right formula. p equals, not Mv but Mv. That's it. Now when you're going to fix up a formula, meaning writing a relativistic generalization, you'd better make sure that your new definition agrees with the old definition, in the limit that velocities are small compared to the speed of light. If gamma is close to 1 - and that's what happens whenever v is close to zero - then sure enough, p is mv and so you do nicely reduce to the old non-relativistic formula. But if something is moving really fast, this formula says the correct momentum will be significantly larger than what you had thought using your classical physics ideas.
This formula, by the way, can be generalized into three dimensions. We won't be doing this very much, but you just have to put a vector sign on the equation. There's no vector sign on top of that gamma. is just a number - the gamma factor. And so, this is all you would need if you really wanted to think about momentum in three dimensions.
Let me think about this formula, p = mv. Let me draw a picture - a graph - of momentum versus velocity for some particle. If the particle's at rest, it's got zero momentum, and it's the simplest function in the world. It's just a straight line. If you double the speed, you double the momentum. It's a linear function. Now what does relativity say? Relativity says t = mv. So for small velocities, you get = 1 x mv. So you're going to get exactly the same prediction, according the theory of relativity, until v starts getting close to the speed of light. When you're going at relativistic speeds, you take the old formula, and you multiply it by something bigger than 1, and so your curve does this. And in fact, if v was ever able to reach c, gamma goes to infinity. You would have a particle with an infinite amount of momentum, which is physically impossible. So nothing ever ends up going at the speed of light. No physical object ever goes at the speed of light. No objects with rest mass of m[0] can ever get up to the speed of light.
And look at this formula. Think about what it's telling you about forces. I haven't told you Newton's second law, yet, in the relativistic form. Remember Newton's second law: F = mA. And now you're wondering, "Do I stick a gamma in front?" Well, the correct formula to use is F is dpdt. This is the correct relativistic generalization of Newton's second law. And it says that as objects get close to the speed of light and the momentum is approaching infinity, a change in the momentum, if you want to speed them up, would be infinitely large and you would require an infinite force. So it's physically impossible to take some chunk of material and speed it up all the way to the speed of light or higher. Objects can't go that fast. They can get arbitrarily close, if you've got enough force pushing on them.
Let's look at some numbers. Let me think about an electron, which is a small particle, so we might have a chance of getting such a small object going fast. And let me try to get it up to a v final of 99 percent the speed of light. This is actually not such a difficult challenge. People do this all the time in physics experiments. In fact, we can get electrons going at 99.99999 percent the speed of light. And how much average force would it take? Average force would be . So let's say we want to do this in a certain amount of time, like 1 second. So we'd have . p final would be .99 times c - that's the velocity times the mass. And I'm dividing by t and I'm starting from zero. And this is the non-relativistic formula. And the relativistic formula says - don't forget I have to multiply by gamma. So you've got to figure out what's gamma. And gamma, when v is .99c - just plug in ; you get about 7 x m[0]c - not approximately 1 x m[0] x c, but 7 times more divided by 1 second minus zero. It takes seven times more average force to get an electron up to 99 percent the speed of light than you would have thought if you'd used Newtonian ideas.
So if you're building a physics accelerator, you really have to know about relativity. The accelerator will fail miserably because you won't supply enough force by huge factors - factors of seven or more. It depends on how fast you want the particle to go. You really need to know about relativity. And there's all sorts of applications where it's important. These effects can become very large as soon as speeds get close to the speed of light.
Conservation of momentum is absolutely correct. It's still one of the cornerstones of physics, even when things are traveling close to the speed of light and smashing into one another, as long as you use this new modified formula for momentum. And it is a strange idea that momentum is growing up to infinity as objects get close to the speed of light, but it's correct and it's been experimentally observed many, many times.
Let me tell you one more thing about notation. Just so you know, when I write p = m[0]v, some people really would like to write this formula in the form mv. They would like to say momentum is mass times velocity. And so they define relativistic mass. They will say relativistic mass m is defined to be x m[0]. m[0] is the mass that the object has when it's at rest, when gamma is 1. And then, as you make the thing go faster and faster as it gets more and more momentum, some people would say its received mass is increasing and it goes up to infinity as the object approaches the speed of light. That's okay. It's semantics. I prefer to think of the mass of an object as m[0]. Some people will think about the relativistic mass. You just have to be careful about what notation you're using and keep it straight. You cannot just say F = mA and use the relativistic mass. That will fail. So I prefer to always think of mass as the rest mass and then I stick in the gamma factor explicitly to figure out the momentum.
Relativity
Relativistic Dynamics
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