Special Relativity Lecture

Transcription

1 Special Relativity Lecture Prateek Puri August 24, 200 The Lorentz Transformation So far, we have proved length contraction and time dilation only for very specific situations involving light clocks; however, we must derive these phenomena more generally if we wish to apply our results extensively. The Lorentz Transformation does this by generally describing the relationship between the time and space measurements of two observers in relative motion with each other. After deriving this transformation, we will be able to convert the coordinates of one reference frame into the coordinates of a different reference frame and notice any length or time distortions. To begin our derivation, we must consider two reference frames that are in relative motion with each other as in Figure 4. Figure : (a) Two reference frames in relative motion with each other We now wish to describe the time and space coordinates of one reference frame with respect to the other. If we allow x to be moving with a certain velocity v 0 with respect to x, we can say x = ax + bt () This is true because the coordinates of the stationary reference frame will be equal to the initial x coordinate of the moving frame plus the amount of distance the moving reference frame has traveled in a certain time, or bv 0. You may be curious about the coefficients a and b, and these were included in anticipation of the space and time distortions we expect to see develop. We must now describe the coordinates of the moving reference frame with respect to the stationary reference frame. Since the moving reference frame is

2 moving away from the stationary reference frame, its coordinates will equal the initial x coordinate of the stationary reference frame minus the distance the moving frame has traveled in a certain amount of time, or x = ax bt (2) We now wish to find the relative velocity between the two reference frames, and to do this, allow x to equal zero x = 0 = ax bt (3) ax = bt (4) v 0 = x t = b a (5) Now allow a light signal to travel in the positive x direction. The position of the light signal can be described by the following simple equations in each respective reference frame x = ct x = ct (6) Substituting these equations into () and (2), we get ct = act + bt ct = act bt (7) Now we will solve for t ct = (ac + b)t ct = (ac b)t (8) t = ct ac + b Let us now substitute our result into the right side of (8) ( ) c t c = (ac b) t (0) ac + b equation = (ac b)(ac + b) (9) = a 2 b 2 (2) Now since v 0 = b a, b = av 0. Substituting this into (2), = a 2 a 2 v 2 0 (3) = a 2 ( v 2 0) (4) 2

3 Solving for a, c2 a 2 = v0 2 = ( ) (5) v2 0 a = ( ) /2 (6) v2 0 Now that we have solved for a, we can also solve for b by using the relation b = v 0 a b = v 0 a = v 0 ( v2 0 ) /2 (7) Since we have now have expressions for a and b, we can substitute into () and (2) x = ax + bt = ( ) /2 (x + v 0 t ) (8) v2 0 x = ax bt = ( ) /2 (x v 0 t) (9) v2 0 Given (8) and (9), we can also solve for t and t by using simple elementary algebra. Doing so, we receive t = ( ) /2 (t + v 0 x / ) (20) v2 0 t = ( ) /2 (t v 0 x/c 2 ) (2) v2 0 Thus we now have derived the Lorentz Transformations and have a system of equations that will allow us to convert coordinates between reference frames in relative motion with each other. Also, we can see time dilation and length contraction flow naturally out of these equations. Allow two points, x and x 2, to denote the beginning and ending points of an object in a certain reference frame. The total length of the object must be x 2 x = l 0 (22) Now we wish to find the length of this object in a reference frame that is moving with a certain velocity with respect to our initial frame. We will use the same procedure as in the previous reference frame and simply calculate the difference between the starting and ending points of the object, hence 3

4 However, from (8) we know that x = x 2 = x 2 x = l 0 (23) ( ) /2 (x + v 0 t ) (24) v2 0 ( ) /2 (x2 + v 0 t ) (25) v2 0 And using the fact that x 2 x = l 0, we can also see Rearranging terms, l 0 = x 2 x ( v2 0 ) /2 (26) l = ( ) /2 (27) v2 0 l = l 0 ( v2 0 ) /2 (28) We have now successfully derived relativistic length contraction in the general case. Next, we will turn our attention to time dilation. This derivation will be very similar to our length contraction, and will begin by denoting the starting and ending times of a certain event as t and t 2. Let us now calculate the time coordinates of these events with respect to a moving reference frame. From (2), we can see that t = ( ) /2 (t v 0 x/c 2 ) (29) v2 0 t 2 = ( ) /2 (t 2 v 0 x/c 2 ) (30) v2 0 We are interested in the elapsed time of an event, and thus we must calculate t 2 t 2. Doing so, we see t 2 t 2 = ( ) /2 (t 2 t ) (3) v2 0 which is the same as the time dilation formula we calculated earlier. 4

5 2 The Spacetime Invariant So far, special relativity appears to have constructed a very subjective image of the universe. Space and time are relative, there is no absolute reference frame, and observers are capable of having unique yet equally valid perceptions of the world. Basically, reality itself is not objective but is dependent on the state of the observer. However, despite all of this subjectivity, is there a common ground between all observers? Is there a quantity that can unify different observers measurements of the world? The answer is yes, and such a quantity is known as the spacetime invariant. The spacetime invariant is the quantity (ct) 2 x 2, and this value is constant for a certain event and is independent of the reference frame used to describe the event. For an example, let us say that a certain event has space and time coordinates of (x, t). Then (ct ) 2 x 2 = ( )[(ct vx/c) 2 (x vt) 2 ] v2 After factoring out a t and an x, we get = ( )[( v 2 )t 2 ( v 2 / )x 2 ] v2 = ( )[( ( v2 ))t2 ( v 2 / )x 2 ] v2 = ct 2 x 2 Thus we have proved the following relation ct 2 x 2 = (ct ) 2 x 2 and we have shown that the spacetime invariant does indeed yield a constant value for a particular event, regardless of the coordinate system used to measure that event. Some may still be curious as to what exactly the spacetime invariant represents. While the physical meaning of the spacetime invariant is not immediately obvious, it is most easily understood as the distance in spacetime, the fourdimensional fabric in which all events take place. 3 Conclusion and Further Study In summary, special relativity can be analogized to opinions. Every person may have a unique opinion on the world that seems completely justifiable to him; however, another person may have a completely different opinion that seems just as valid. The two may foolishly argue over whose opinion is actually right, but in the end opinions are opinions, and there is no way to determine what is actually 5

6 fact. In the same way, perspectives of the world are completely dependent on state of the observer and there is no way to determine an absolute reference frame (unless you are an ether theorist of course). Using these ideas, Albert Einstein radically reshaped the way we viewed and understood the world with his theory of special relativity, and I hope this paper has effectively explained some of the theory s core concepts. However, there is still much to learn about special relativity, and I strongly encourage those interested in relativity to continue to explore the subject. There are still many interesting topics to discuss, such as the Twin paradox, relativistic mass increase, causality, and more. 4 Sources (for figures)