Hi everyone. Welcome to today’s episode of your daily equation. And today I’m going to focus upon an equation that I feel doesn’t get enough airtime when people talk about the strangeness of space and time and relativity. Because it’s an equation that directly addresses the kind of question that I at least am asked all the time by people who encounter these strange ideas, especially the idea of the constant nature of the speed of light.
Because, look, we all have, in our ingrained intuition, the following fact, right? If you run toward an object that’s approaching you, it will approach you faster. And if you run away from an object that’s approaching you, it will approach you slower, right? And yet we know that that intuition cannot be completely true because if the object that’s approaching you is a beam of light, then that would suggest that by running toward it, you could make the speed of approach faster than the speed of light. And if you’re on a way from the approaching beam, it should make the speed of approach slower. But the constant nature of the speed of light says that that can’t be true. So how do we reconcile these ideas? And today’s rather beautiful and simple mathematical equation will show us how Einstein’s theory copes with this tension and makes complete sense of it.
Okay, so let’s jump right in, and I’ll begin with a little again, silly story that just gets our mind in the right perspective for the ideas that we’re discussing. So what is this story? So imagine that there’s a nice little game of catch happening between George and Gracie. And say George is throwing that football toward Gracie at five minutes per second, then Gracie receives it at five minutes per second, nothing tricky about that.
But now imagine the next day George comes out with not a football but an egg, and Gracie is not fond of playing catch with eggs, so what does she do? She turns and runs because of that intuition, that by running away the speed of approach of the egg will be lessened. It will be made smaller, indeed putting some numbers behind it. If the egg is flying in the horizontal direction toward Gracie at five meters per second, and she runs away, say at three minutes per second, then we all know in our intuition that the egg should be approaching her with a net speed of two meters per second.
And in the reverse situation too, if Gracie loved playing catch with eggs and couldn’t resist the wait for the egg to reach her and she ran toward George at say the same speed, three meters per second, then we all have in our tuition that the egg would approach her at five plus three meters per second or eight meters per second. And the tension then comes in when we think about these ideas applied to the speed of light. So let me show you that, let me bring up my iPad here.
So what’s the basic formula that Gracie and George, and we are making use of? The basic formula is that if an object is approaching you say at V meters per second when you are stationary, and if you run away from it, then if you run at a speed W with respect to the ground, say that initial frame of reference, then V minus W this should be the speed of approach in that circumstance. And the reverse, that I also mentioned, if the object, say the egg is approaching at a speed V and you run toward it with the speed W, then you should have a net speed of approach of V plus W.
And the tension that I’m mentioning just to make it explicit is, what if you don’t have a football, you don’t have an egg but rather you say, have a beam of light. So now the initial speed of approach is C in both of these cases, and if you run away or run toward the beam of light with the speed W, then the speed of approach from this reasoning should be C minus W, which would be of course less than C, or C plus W, if you run toward the beam of light. And that of course is greater than C. And that’s the problem, speeds less than the speed of light is [inaudible 00:04:45] greater than the speed of light when you are encountering a beam of light who’s speed is meant to be constant independent of your motions, how do we make sense of this?
Well, the basic idea that Einstein tells us is that even this very simple formula that we’re all familiar with from elementary physics or even just elementary logic is actually wrong. It works really well at speeds that are much less than the speed of light, and that’s why we all hold it in our intuition. But Einstein actually taught us that each of these formula needs a correction. Let me show you what the correction is, is that is today’s daily equation.
So instead of V minus W, Einstein says that the correct formula of the speed of approach if you’re running away from an object that has speed V, and you’re running way at speed W is corrected by one minus V times W divided by C squared, and the V plus W formula gets a very similar correction. And that correction just has the other sign. In fact, you could do these altogether with one formula that just had the plus sign if you let all the speeds have positive and negative values. But let me just keep it simple and imagine that all the speeds involved are positive, V and W are positive numbers.
So these are the formula. They’re effectively the same formula just with the two cases that we’re writing down separately. And that is the so-called relativistic velocity combination law. And now let me just show you how this works. If for example, you are taking V to be equal to C, now you’re not throwing the egg or the football but you are shining a beam of light. So the case where you run away, Gracie say, runs away from the beam of light, we give a C minus W over one minus C times W over C squared, and what does that equal? Well, look, we can write this as C minus W over one minus W over C, and we can write that as C times. Just pull out a C upstairs, one minus W over C divided by one minus W over C. And now you see that the one minus W over C factor cancels in the top and the bottom, and that then gives us the net result is equal to C and that’s fantastic.
So by running away from the beam of light, Gracie does not decrease the speed of approach of the light, that’s correction factor that Einstein gives us over here, has this wonderful effect of ensuring that the combined velocity is still equal to C. And as you can imagine, and I don’t even need to go through it, I can just put plus signs in here. If Gracie was running toward the beam of light, all the analysis would have a plus there and you’d again have this cancellation, and you get speed of light again as your result if Gracie is running toward the oncoming and beam of light that George shines at her.
Now, that’s the special case where V is equal to C. It’s fun to use this formula even in other circumstances. Imagine that you have an object that is being fired at you, say at three quarters the speed of light, and let’s say you run toward it at three quarters the speed of light, just for the fun of it.
Now, your naive, classical intuition would tell you that the net speed from your perspective would be three quarters the speed of light plus three quarters of the speed of light is coming toward you and you’re running toward it, the speeds would combine, and the intuitive way of doing these kinds of calculations, but of course that number would be six quarters of the speed of light that’s bigger than the speed of light problem. Well, what does Einstein do? He says, “Hang on,” you need to correct this by one plus VW over C squared. VW now is three quarters C times three quarters of C divided by C squared, and now we can work this out.
Upstairs we have the offending six quarters of the speed of light, but what do we get downstairs? Downstairs, we get one plus three quarters times three quarters is nine over 16, and the C square, it’s cancel. So we get six quarters C times, what’s one plus nine over 16? Well, this guy over here just gives us 16 over 16 plus nine over 16, which is 25 over 16, which you can bring that upstairs as 16 over 25. And now the four goes in here and we get 20, and we left out the C, we got 24 over 25 times C less than the speed of light.
So the offensive term, six quarters times the speed of light is reduced by the correction factor to 24 over 25 times the speed of light less than C. And that will always be the case. Whatever numbers you put in to this relativistic velocity combination formula, it will always yield a net speed from your perspective from say, Gracie’s perspective, that is less than the speed of light regardless of the speeds that are put into that formula, as long as each such speed is less than or equal to the speed of light.
So it’s a beautiful formula, and it actually shows us indeed. Just going back to the initial, little scenario that we started with, with George and Gracie, say with the egg. So in that case, in fact, let me just bring this up for the heck of it because it’s fun to see. So in that particular case we had V equals to five, we’re not going to put the units in, and W, say was equal to three. And we did this little calculation that five minus three equals two, I’ll put it in meters per second, meters per second looks funny to me. Otherwise, meters per second, meters per second.
So that was the calculation that we did in everyday life. But Einstein’s telling us, even in everyday life, you need to include this correction. So what is the actual speed of the approaching egg from Gracie’s perspective, while you do five minus three meters per second upstairs, but now you must divide by one minus five meters per second times three meters per second, divided by the speed of light squared, which of course in meters per second is a nice big number, three times ten to be eight meters per second. So what is this correction factor? Well, the correction factor is of course quite small, or I should say, differs from one by a little bit. It’s one minus this really tiny number that we have over here, which C squared is about 10 to the 17. So call this on the order of a correction factor in the 16th decimal place or so, 10 to the minus 16 or so.
So the net effect is that this number two that we have over here is actually increased by a little bit, because you’re dividing through by a number, which is itself less than one. It’s very close to one, only differs from one way down and say the 15th or 16th decimal place, but it is a little bit less than one, which means that this two would be a little bit bigger than two.
So the speed of approach even in everyday life in that simple silly scenario of the egg approaching Gracie and she runs away. Her intuitive calculation is close to correct, but it’s not completely correct. The effects of relativity are always there, they’re just really small, typically at everyday speeds. But they are there, and they matter, and they show us how, when the speeds approach or in fact are equal to the speed of light, everything combines in just the right way to give net speeds that are always less than or equal to the speed of light, just as relativity requires.
Okay. That’s all I had to say for today, this beautiful relativistic velocity combination law that allows us to correct our intuition for how speeds combine, making everything compatible with the speed of light being the maximum speed limit, making the world safe, [inaudible 00:13:49] relativity. Okay. Until next time, take care, this is your daily equation.