 Third party, another object from the perspective of those two frames, and thinking about the velocity of that object as perceived in the two frames. We'll also learn how to properly add velocities of objects to frame velocities in special relativity. Let's use a concrete example to motivate a kind of basic problem we can use going forward to think about the question of object velocities relative to moving frames of reference. So the example I will pick for this is a non-copyright violating space wars. Recently in a globular cluster fairly nearby two ships were engaged in a chase. The lead ship is moving away from the pursuing ship at a velocity given by the vector v. The pursuing ship fires a projectile straight at the lead ship along the line of motion and at a velocity vector u relative to the firing ship. With what velocity does the lead ship observe the projectile to move? Now I've illustrated this with a little graphic cartoon right here. We have the pursuing ship on the left, the projectile it's fired with the velocity of the projectile from its perspective drawn here in red. The ship it's pursuing over here on the right and the velocity of that ship being pursued relative to the pursuing ship given by the vector v. Now in the Galilean or Newtonian view of space and time the answer to the question with what velocity does the lead ship observe the projectile to move is rather straightforward. The observed velocity in its frame of reference u vector prime would be equal to the velocity of that ship with respect to the pursuing ship minus the velocity of the projectile with respect to the pursuing ship. That would also turn out to be completely wrong when the velocities in this question approach velocities near that of light. So for instance if the projectile is actually a beam of light imagine a laser beam a laser cannon mounted on the front of the pursuing ship it turns on the cannon the beam is emitted this is a beam of pure light it should move at the speed of light. We plug that into this calculation we get all kinds of wrong answers here. The lead ship doesn't doesn't see the laser beam approaching at the speed of light and we know that's just not consistent with observation as encoded in the postulates of special relativity. So what then is the correct addition of velocities in a problem like this? And that's the question we want to figure out in this lecture. We can begin by writing down the Lorenz transformation equations treating the pursuing ship as the rest frame the lead ship as the moving frame and the projectile as an object to be located or studied in either frame. The space-time coordinates of that object in each frame are given as follows for example if we have the space-time coordinates x and t in the rest frame we can get the space-time coordinates in the moving frame the s prime frame using this version of the Lorenz transformation equations yielding x prime and t prime the location and the time at which the location is observed for the projectile in the perspective of the lead ship. Now we can write differentials of space and time using calculus dx prime and dt prime and this will allow us to work toward obtaining equations with velocities. So for instance u prime is the first derivative of x prime with respect to t prime after all that would be the velocity of the object as observed in the lead ship or moving frame u would be the first derivative of x with respect to t that's the perspective of the projectiles velocity from the rest frame or the pursuing ship. Now if this particular step feels weirdly familiar to you In an earlier lecture I walked you through a brief example as to why the Lorenz transformation needs to be a linear transformation between moving frame coordinates and rest frame coordinates and we came dangerously close in that lecture to deriving the velocity transform albeit I was doing that for arbitrary powers x to the n and t to the m for instance here of course it's purely linear because it's based on the Lorenz transformation and so if some of this feels awkwardly familiar you may flip back to the earlier lecture on the Lorenz transformation and have a look and see where the roots of this were planted. So the differential of space in the moving frame dx prime is going to be equal to gamma times the quantity dx minus v dt and the differential of time in the moving frame is going to be equal to gamma times the quantity negative v over c square dx plus dt. Now we can take the ratio of dx prime over dt prime and this allows us to get the velocity u prime of the projectile as observed in the moving frame or the frame of the lead ship substituting in with our differentials for dx prime and dt prime we arrive at this rather unpleasant looking equation but one of the nice things about this is that the leading gamma factors the one over square root of one minus v squared over c squared terms they cancel out in both the numerator and the denominator and this leaves us with an equation that looks like this just in terms of the remaining differentials of dx and dt now if we divide the top and the bottom by dt the little unit of time that we're considering then we wind up with terms that go like dx over dt which is just u the velocity of the projectile in this case entirely along the x axis and so this equation takes the following form which at the end of things doesn't look horrible the velocity of the projectile as observed in the moving frame the frame of the lead ship is simply given by the velocity of the projectile as launched from the perspective of the rest frame the pursuing ship minus the velocity of the frame so the velocity difference between the lead ship and the pursuing ship divided by a quantity that goes like the motion one minus u v over c squared so we have arrived at a formula for combining the velocities of the moving frame with the velocity of the projectile as observed in the rest frame to allow us to compute the observed velocity of the projectile in the moving frame this equation is a substitution for the old Galilean transformation addition of velocities equation and is correct from the perspective of special relativity so let's plug in some numbers and see what we learn about projectile motion in the case where objects are also in relative motion to each other and observing that projectile as it moves and let's begin by picking a low velocity situation where the ships are not really moving apart from each other all that fast i've decided to pick the lead ship having a velocity of just one percent the speed of light or 0.01 c and i've picked a projectile velocity that's just three times bigger than that or three percent the speed of light 0.03 c from the perspective of the firing ship the pursuing ship now from the above equation we learn that the lead ship observes the projectile approaching it at a speed of 0.02 c now if you stare at this for a moment and recall the Galilean velocity transformation you'll note that this is exactly what we would have expected from the low speed case where all the velocities of objects in the problem are not really a large fraction of the speed of light although i've allowed them in this case to go up to a few percent the speed of light we actually get back exactly what would have been told to us by the velocity transformation in Newtonian slash Galilean relativity that is that u prime equals u minus v now that doesn't mean that this is exactly true at every decimal place there's some decimal place where the Newtonian Galilean approximation to space and time in motion breaks down compared to the more accurate special relativistic calculation so let's instead pick some bigger velocities let's now assume that the lead ship is racing away from the pursuing ship at half the speed of light and then from the perspective of the pursuing ship it fires this projectile at eight tenths the speed of light 0.8 c plugging those numbers in we find out that the lead ship observes the projectile to approach it at one half the speed of light and if you stare at that again for a moment play around with the numbers on your own you'll very quickly realize that this is definitely not what would have been predicted by the Newtonian or Galilean approach it's not simply u minus v in this case now interestingly we can look at the case of when the lead ship is flying toward the pursuer so now we turn the lead ship around and we aim it back at the pursuing ship and flip its velocity vector so that it's moving at negative 0.5 c from compared to its original direction of motion in that case we see that the lead ship that's now racing toward the projectile that's been fired at it doesn't observe that projectile to be moving in excess of the speed of light rather it observes it to be moving at 93 percent the speed of light and that's again a distinction from what the Newtonian or Galilean approach would have yielded the old relativistic approach from Galilean relativity would have predicted that the lead ship observes the projectile to be approaching at a speed that is far in excess of the speed of light but we also know from the postulates of special relativity that one consequence is that nothing can move faster than the speed of light and so we see that that's preserved here in the velocity transformation although the velocity of the ship is now aimed back at its pursuer and although the naive addition of velocities would give you something in excess of the speed of light the naive addition is not consistent with observations of space and time and the speed of light and using the special relativistic transformation we see that while it's true that the velocity of the projectile does appear to be larger than when the lead ship was racing away from it it does not exceed the speed of light but comes in at a pretty pretty fair fraction of the speed of light so let's summarize what we've learned about adding velocities in special relativity keeping in mind that the cases that I've built these equations from all involved an object velocity that was parallel or anti-parallel to the velocity of the frames if you have the velocity of the object in the rest frame and want to determine it in the moving frame then the left equation is what you want if on the other hand you have the object velocity in the moving frame and you want to determine it in the rest frame all that should change between the left equation and its corresponding equation on the right should be that you swap u and u prime and you flip the sign of all terms that involve v or v cubed or something like that you take v and send it to negative v and in fact that's the equation that's written here on the right you can always derive these directly from the Lorenz transformation or you can memorize one of them and remember how to transform it into the other by swapping the object velocities and flipping the sign of the frame velocities I'll leave it up to you as to what your best possible learning strategy is for this but know that if you memorize one of these you can figure out the other from context and knowing how to trade the mathematics around now what if the object instead of having its velocity aligned parallel or anti-parallel to the frame velocities is moving in a direction that isn't solely parallel or anti-parallel to v so you might be tempted to assume that the object velocity in for instance the y direction assuming that the frames are moving only along the x and x prime axes you might be tempted to assume that the object velocity along the y direction and the z direction as observed in either frame is the same since in the Lorenz transformation coordinates y and y prime z and z prime are equal to each other if all the motion is along x and x prime and you'd be wrong you need to be very careful with these things why well think about it a second object velocity necessarily involves the time derivative of a coordinate is time absolute between two different frames of reference well we should feel pretty confident at this point that the answer is that it does not t does not equal t prime in special relativity because a time derivative is involved there's going to be a dy dt and there's going to be a dy prime dt prime and while y may be equal to y prime t is not equal to t prime so let's go through this i'm going to consider motion component along the y axis the frames are moving entirely along x so v and this is still directed entirely along the x and x prime axes but i'm going to allow the velocity of the object to develop a component u y or u y prime along the y and y prime axes respectively so let's look at what the transformation of u prime to u would be for the case of this component along the y axis and y prime axis so we know that in the rest frame u subscript y is just dy dt it's the change in the y coordinate with respect to time as observed in the rest frame now it's true that in the Lorenz transformation if the motion is entirely along x and x prime that y does equal y prime so we can replace dy with dy prime and no harm no foul that's mathematically allowed but if we're going to substitute for dt with dt prime we have to use the full glory of the differential form of the time equation in the Lorenz transformation and that means replacing dt with the quantity i show here in the denominator of this fraction now of course i can divide the top in the bottom by dt so that i get a u y prime in the numerator in the denominator gammas don't cancel out in this case though between the numerator and the denominator because y equals y prime y and y prime don't depend on a gamma factor to correct between them and as a result it's actually an easier derivation i feel than for the case of the object motion component along the direction of travel of the the frame relative to the rest frame but it's not perhaps quite as a memorable looking now similarly if we have u y prime and want u y all we have to do is swap u y prime and u y in these equations and replace v with minus v and so the corresponding equation that tells us what the velocity component in the moving frame looks like given the velocity component in the rest frame will be the one i show here and by the way if there's a component of motion along z and z prime you can obtain a similar equation it has pretty much exactly the same form as the one shown here with u y replaced by u z and u y prime replaced by u z prime um you can very quickly write that equation down but i just want to go through this because it's important to recognize that while it's true that y equals y prime and z equals z prime when the motion is entirely along x and x prime it is not true that u y is naively equal to u y prime and that's because a time derivative is involved and time does not pass the same way in the two frames when one is moving relative to the other finally let's take a look at one last special case and that is if the pursuing ship shoots a laser beam at the lead ship so what i've done is i've replaced the red projectile with a red squiggly line to represent an electromagnetic wave light being fired at the lead ship now the lead ship is still moving at a velocity v vector with respect to the pursuing ship i've put everything along the horizontal axis here but now the velocity of the projectile is c because this is a beam of light and so it will always and forever move at exactly the speed of light so the speed of this projectile is now exactly 2.998 times 10 to the eighth meters per second as viewed from all frames so if the pursuing ship had fired a weapon like this a laser beam a beam of light well we know that the second postulate of special relativity demands that all observers must see light moving at sea regardless of their state of motion so does this velocity addition relationship capture that postulate in all of its full glory well let's find out let's assume that the relative velocity of the lead ship to the pursuing ship is one half c and that the projectile speed as viewed in the rest frame of the pursuing ship is c the speed of light well plugging these numbers in we can start from the equation where we have the relative velocity of the two frames and the speed of the projectile in the rest frame and we can get the speed of the projectile as observed in the moving frame so all i've done is i've replaced u in this equation with c because the projectile is a speed up is a beam of light that's moving at the speed of light if you do some algebra you can simplify this equation to c minus v all over the quantity one minus v over c and if you do a little bit more algebra you'll find out that this is just equal to c minus v over the quantity one over c times c minus v and you play with this one step further you find out that this is just equal to c the speed of light so in fact we see that v entirely drops out of this equation once the projectile is a light beam the value of v doesn't matter at all the relative velocity between these two vessels can be any number and it won't affect the outcome of the calculation v could have been at a half c or negative a half c or point eight c or point nine nine nine nine c basically once u equals c v drops entirely out of the equation and we always recover that u prime equals c as well the second postulate of special relativity is fully obeyed by this velocity transformation equation so to review in this lecture we have learned how to think about object velocities in different frames of reference and how to go from the coordinates of an object that's in motion to its velocity in different frames we've then used that information to figure out how to properly add velocities of objects to frame velocities in special relativity we've looked at a couple of case studies of this and seen that everything seems to comport with the postulates of special relativity which themselves comport with observations of the natural world