From Special Relativity to Feynman Diagrams.pdf

# When we shall consider four dimensional spacetime

This preview shows pages 134–136. Sign up to view the full content.

When we shall consider four-dimensional space–time instead of the three- dimensional Euclidean space, among all the possible transformations on a RF, of particular interest are the Lorentz transformations, on which Einstein’s principle of relativity is based. We shall show, at the end of this chapter, that Lorentz transforma- tions close a group, the Lorentz group . If we also include space–time translations, this group enlarges to the Poincaré group . If physical laws are expressed as an equal- ity between tensors of the same type with respect to the Lorentz group, we will be guaranteed that the principle of relativity holds. 4.7 Minkowski Space–Time and Lorentz Transformations In discussing special relativity, we have seen that space–time can be regarded as a four-dimensional space M 4 whose points are described by a set of four Cartesian coordinates ( x μ ) = ( x 0 , x 1 , x 2 , x 3 ), μ = 0 , 1 , 2 , 3 , (4.140) three of which ( x i ) = ( x 1 , x 2 , x 3 ) = ( x , y , z ) are spatial coordinates of our Euclid- ean space E 3 , and one x 0 = ct is related to time. A point on M 4 describes an event taking place at the point ( x , y , z ) , at the time t . Just as for Euclidean space, we can

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4.7 Minkowski Space–Time and Lorentz Transformations 123 define vectors connecting couples of points in M 4 , like the infinitesimal displacement vector connecting two infinitely close events: dx ( dx μ ) = ( dx 0 , dx 1 , dx 2 , dx 3 ). (4.141) These vectors span a four-dimensional linear vector space on which a symmetric scalar product is defined by means of the metric g μν = η μν , where 12 : η μν = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 . (4.142) Given two 4-vectors P ( P μ ) and Q ( Q μ ) , their scalar product reads: P · Q = P μ η μν Q ν = P 0 Q 0 3 i = 1 P i Q i . (4.143) This scalar product, in contrast to the one defined on the Euclidean space, is not positivedefinite,namelydoesnotsatisfyproperty( c )of( 4.7 ),sincethecorresponding metrichasonepositiveandthreenegativediagonalentries(indefiniteorMinkowskian signature). As a consequence of this the squared norm of a 4-vector P ( P μ ) , defined using this scalar product: P 2 P · P P μ η μν P ν = ( P 0 ) 2 3 i = 1 ( P i ) 2 , (4.144) can vanish even if P is not zero. In particular a non-vanishing 4-vector can have positive, zero or negative squared norm, in which cases we talk about a time-like, null or space-like 4-vector, respectively. We can take, as 4-vector, the displacement vector dx , whose squared norm measures the squared space–time distance ds 2 between two infinitely close events: ds 2 = dx 2 = dx μ η μν dx ν = ( dx 0 ) 2 ( dx 1 ) 2 ( dx 2 ) 2 ( dx 3 ) 3 . (4.145) As pointed out when discussing about relativity, the distance ds in ( 4.145 ) should be interpreted as the infinitesimal proper-time interval times the velocity of light: ds 2 = c 2 d τ 2 . A four-dimensional space on which the metric ( 4.142 ) is defined, is called Minkowski space (or better space–time).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern