MISN-0-75 RELATIVISTIC SPACE-TIME: FOUR-VECTORS RELATIVISTIC SPACE-TIME: FOUR-VECTORS by ct Frank Zerilli an event 1. Description (popping of a flashbulb) a. Events in Spacetime . 1 b. Four-vectors . .1 c. World Lines . 1 y 2. The Lorentz Transformation as a Matrix . 2 3. Precise De¯nition of a 4-Vector . 4 x 4. Proof of the Momentum-Energy 4-Vector . 5 Acknowledgments. .8 Project PHYSNET·Physics Bldg.·Michigan State University·East Lansing, MI 1 2 ID Sheet: MISN-0-75 THIS IS A DEVELOPMENTAL-STAGE PUBLICATION Title: Relativistic Space-Time: Four-Vectors OF PROJECT PHYSNET Author: Frank Zerilli, Michigan State University The goal of our project is to assist a network of educators and scientists in Version: 2/1/2000 Evaluation: Stage 0 transferring physics from one person to another. We support manuscript processing and distribution, along with communication and information Length: 1 hr; 20 pages systems. We also work with employers to identify basic scienti¯c skills Input Skills: as well as physics topics that are needed in science and technology. A number of our publications are aimed at assisting users in acquiring such 1. Write a given system of linear equations in matrix form (MISN-0- skills. 301). 2. State down the Lorentz transformation and use it to transform Our publications are designed: (i) to be updated quickly in response to given positions and times (MISN-0-12). ¯eld tests and new scienti¯c developments; (ii) to be used in both class- 3. De¯ne relativistic energy and momentum in terms of mass and room and professional settings; (iii) to show the prerequisite dependen- velocity (MISN-0-24). cies existing among the various chunks of physics knowledge and skill, as a guide both to mental organization and to use of the materials; and Output Skills (Knowledge): (iv) to be adapted quickly to speci¯c user needs ranging from single-skill K1. Vocabulary: event, four-vector, world line. instruction to complete custom textbooks. K2. Write the Lorentz transformation in matrix form. New authors, reviewers and ¯eld testers are welcome. K3. Show that relativistic energy and momentum are the components of a four-vector. PROJECT STAFF Output Skills (Problem Solving): Andrew Schnepp Webmaster S1. Transform the coordinates of a given event from one inertial frame Eugene Kales Graphics to another. Peter Signell Project Director S2. Transform given energy-momentum four-vectors from one given Lorentz (inertial) frame to another. ADVISORY COMMITTEE Post-Options: D. Alan Bromley Yale University 1. \Topics in Relativity: Doppler E®ect and Pair Production" E. Leonard Jossem The Ohio State University (MISN-0-308). A. A. Strassenburg S. U. N. Y., Stony Brook Views expressed in a module are those of the module author(s) and are not necessarily those of other project participants. c 2001, Peter Signell for Project PHYSNET, Physics-Astronomy Bldg., Mic° h. State Univ., E. Lansing, MI 48824; (517) 355-3784. For our liberal use policies see: http://www.physnet.org/home/modules/license.html. 3 4 MISN-0-75 1 MISN-0-75 2 RELATIVISTIC SPACE-TIME: FOUR-VECTORS ct world line of an by world line of a arbitrarily moving point Frank Zerilli point object stationary at (x0,y0) 1. Description y0 y 1a. Events in Spacetime. The term \event" is used to specify a x0 point in what we will call spacetime which consists of the usual three- x dimensional euclidean space of location plus the one-dimensional axis of time. As an example, suppose we ¯re a flashbulb (see Fig. 1). The ¯r- Figure 2. \World lines" in spacetime. ing of the flashbulb is an event and to specify its location in spacetime we must give four quantities - three quantities (for example, rectangular y(t) = y0 coordinates x, y, z in some inertial frame) which specify its location in z(t) = z space, plus one quantity which speci¯es its \location" in time (in some 0 inertial frame). ct = ct 1b. Four-vectors. The four quantities constitute a vector called a where x0, y0, z0 are the space coordinates of the stationary point object. \four-vector" in this four-dimensional space. We will, for convenience, Or, we can say that the four-vector of the point object at any time t is: choose to use (ct) as the fourth component of the four-vector rather than t, since the equations we will write are then a little simpler. Also, the (x0; y0; z0; ct) four components of the vector then all have the same dimension: length. In general, a world line can be speci¯ed by giving a four-vector which is Note: (ct) is the distance light travels in time t. a function of time, or some other parameter, call it ¿: 1c. World Lines. The trajectory of a point object in spacetime is called its \world line." For example, the world line of the point object [ x(¿); y(¿); z(¿); ct(¿) ]: stationary in the frame described by coordinates (x, y, z, ct) is a straight All physical objects have some extension: we speak of the length, width, line parallel to the t-axis and could be described by the set of equations: and height of objects. Upon reflection we see that for a complete speci¯- cation of the extension of an object we should specify not only its spatial x(t) = x0 dimensions but also its extension in time. For example, we might specify the dimension of the building as: 100 ft long, 50 ft wide, 30 ft high, and ct 20 years extended in time (it was built in 1910 and demolished in 1930). an event In a spacetime diagram, we might describe the object by giving the world (popping of a flashbulb) lines for each point of the object (see Fig. 3). y 2. The Lorentz Transformation as a Matrix We know that the space and time coordinates of a point with respect to one Lorentz (inertial) frame are related to the coordinates with respect x to a second Lorentz (inertial) frame by Lorentz transformation. Let L Figure 1. An \event" in spacetime. be a Lorentz frame with coordinates x, y, z, ct and let L0 be a Lorentz 5 6 MISN-0-75 3 MISN-0-75 4 ct x x' L L' y z z' y y' x Figure 4. Two inertial frames and a relative velocity. Figure 3. A collection of world lines for an extended object. 3. Precise De¯nition of a 4-Vector frame with coordinate axes x0, y0, z0, ct0 moving uniformly with velocity v parallel to the z-axis of L. Also let the x0, y0, z0 axes of L0 be parallel to Any set of four quantities which transforms from one Lorentz frame the x, y, z axes, respectively, of L. Then the Lorentz transformation may to another just like the coordinates (x, y, z, ct) of a point in spacetime is be written: called a four-vector.1 Thus (x, y, z, ct) is a four-vector. x0 = x One example of a very important four-vector is the energy- y0 = y momentum four-vector whose spatial components are the three compo- nents of relativistic momentum and whose fourth component is the rela- z0 = γz ¯γct ¡ tivistic energy (divided by c): ct0 = ¯γz + γct ¡ E p ; p ; p ; : where we have utilized the widely used notation: x y z c µ ¶ v ¯ ; γ (1 ¯2)¡1=2: ´ c ´ ¡ Knowing that the components of momentum and energy transform like a four-vector, we can ¯nd the momentum and energy of an object as This may be written as a matrix equation: viewed in any Lorentz frame if we are given the values in any one Lorentz frame. For example, suppose we have a particle of mass m at rest in one 0 x 1 0 0 0 x Lorentz frame. Consider a Lorentz frame moving with velocity v along y0 0 1 0 0 y 0 1 = 0 1 0 1 the x-direction. Then, in this new frame, the particle is moving with z0 0 0 γ ¯γ ¢ z 0 ¡ velocity v along x -axis and: B ct0 C B 0 0 ¯γ γ C B ct C ¡ B C B ¡ C B C @ A @ A @ A 0 px γ ¯γ 0 ¯γm0c 0 = ¡ = ¡ Since the x and y coordinates are not a®ected in this case we may suppress E =c ¯γ γ ¢ m0c γm0c them and write: µ ¶ µ ¡ ¶ µ ¶ µ ¶ So: 0 2 z γ ¯γ z 0 m0v 0 m0c 0 = ¡ px = and E = : ct ¯γ γ ¢ ct ¡(1 v2=c2)1=2 (1 v2=c2)1=2 µ ¶ µ ¡ ¶ µ ¶ ¡ ¡ 1This de¯nition is correct only for flat spacetime. In non-flat spacetimes, coordi- nates are not the components of four-vectors. 7 8 MISN-0-75 5 MISN-0-75 6 1 uz 1 uz = v_ u0 = = x b z uxv ux x' c γ 1 γ 1 ¯ ` ¡ c2 ¡ c u g = 1 m0 Ö1 - b2 The quantity 1 (u0=c)2) occurs several times in the energy and momentum formulas so¡let us calculate it in terms of the velocity in the ` p v ¯rst (unprimed) frame: z z' 0 2 0 2 0 2 0 2 (u ) = (ux) + (uy) + (uz) 2 ¡2 2 2 y y' (ux ¯c) + γ (uy + uz) = ¡ 2 Figure 5. A velocity as seen from a stationary frame and ux 1 ¯ from a moving frame. ¡ c ³ ´ So: This gives us just the expected values for momentum and energy of a particle of mass m0 moving with velocity v. 2 u2 + u2 ¡ ux ¡2 y z 2 ¯ + γ 2 u0 c ¡ Ã c ! 1 = 1 ³ ´ 2 4.