In Newtonian mechanics, we have the space coordinates, e.g.(x,y,z)if we consider a 3-dimensional space with a system of Cartesian coordinates, and an absolute time, t. We have thus “space” and “time” as two distinct entities. In special relativity it is useful to introduce the concept ofspacetime, where the space coordinates and the time coordinate become the coordinates of the spacetime. Every point of the spacetime is anevent, because it indeed represents an “event” occurring at a particular point of the space and at a certain time.
2.2 Minkowski Spacetime 31 First, we want to find the counterpart of the line elementdlin Eq. (1.1). Indeed, if we assume the Einstein Principle of Relativity,dlcannot be an invariant any longer:
if we consider a light signal moving from the pointAto the pointBin two different reference frames, the distance between the two points cannot be the same in the two different reference frames because the speed of light is the same but the travel time is not, in general.
Let us consider the trajectory of a photon in a (3+1)-dimensional spacetime (3 spatial dimensions+1 temporal dimension). If the trajectory is parametrized by the coordinate timet, we can write1
||xμ|| =
⎛
⎜⎜
⎝ x0(t) x1(t) x2(t) x3(t)
⎞
⎟⎟
⎠=
⎛
⎜⎜
⎝ ct x(t) y(t) z(t)
⎞
⎟⎟
⎠= ct
x(t) . (2.2)
Note that we have definedx0 =ct, wherecis the speed of light, andx0thus has the dimensions of a length same as the space coordinatesx1,x2, andx3. Since the speed of light iscin all inertial reference frames, the quantity
ds2 = −c2dt2+d x2+d y2+d z2= −c2dt2+dx2
=
−c2+ d x
dt
2
+ d y
dt
2
+ d z
dt
2
dt2 (2.3)
must vanish along the photon trajectory in all inertial reference frames (the case of non-inertial reference frame will be discussed later;ds2 =0 still holds, but the expression ofds2is different).dsis the line element of the spacetime.
Ifds2=0 in an inertial reference frame, it vanishes in all inertial reference frames.
However, this is not yet enough to say it is an invariant. Indeed we may have
ds2 =k ds2, (2.4)
whereds2is the line element of another inertial reference frame andksome coef- ficient. If we believe that the spacetime is homogeneous (no preferred points) and isotropic (no preferred directions),k cannot depend on the spacetime coordinates xμ. So it can at most be a function of the relative velocity between the two reference frames. However, this is not the case. Let us consider reference frames 0, 1, and 2, and letv01 andv02 be the velocities of, respectively, coordinate systems 1 and 2 relative to coordinate system 0. We can write
ds02=k(v01)ds12, ds02=k(v02)ds22, (2.5)
1Some authors use a different convention. For an (n+1)-dimensional spacetime (nspatial dimen- sions+1 temporal dimension), they write the coordinates of the spacetime as(x1,x2, . . . ,xn+1), wherexn+1is the time coordinate. In such a case, the Minkowski metric in Eq. (2.9) becomes ημν=diag(1,1, . . . ,1,−1).
wherev01andv02are the absolute velocities ofv01andv02becausekcannot depend on their direction given the isotropy of the spacetime. Letv12 be the velocity of reference frame 2 relative to reference frame 1. We have
ds12=k(v12)ds22. (2.6) Combining Eqs. (2.5) and (2.6), we have
k(v12)= ds12
ds22 = k(v02)
k(v01). (2.7)
However, v12 depends on the directions ofv01 andv02, while the right hand side in (2.7) does not. Equation (2.7) thus implies that k is a constant and since it is independent of the reference frame it must be 1. We can now conclude thatds2is an invariant.
The line element (2.3) can be rewritten as
ds2=ημνd xμd xν, (2.8)
whereημν is theMinkowski metricand reads2
||ημν|| =
⎛
⎜⎜
⎝
−1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
⎞
⎟⎟
⎠. (2.9)
Then-dimensionalMinkowski spacetimecan be identified withRnwith the metric ημν.3It is the counterpart of then-dimensional Euclidean space with the metricδi j
reviewed in Sect.1.2. The Minkowski spacetime is the space of relativistic mechanics as the Euclidean space is the space of Newtonian mechanics. In general, we can write the metric of the spacetime asgμν. For example, in spherical coordinates we have
2The convention of a metric with signature(− + ++)is common in the gravity community. In the particle physics community it is more common the convention of a metric with signature(+ − −−).
3Throughout the book, we use Latin lettersi,j,k, . . .for space indices (1,2, . . . ,n), wherenis the number of spatial dimensions, and Greek lettersμ, ν, ρ, . . .for spacetimes indices (0,1,2, . . . ,n).
Such a convention is also used when we sum over repeated indices. For instance, forn=3 we have ds2=ημνd xμd xν≡ −
d x02
+ d x12
+ d x22
+ d x32
. (2.10)
If we wroteηi jd xjd xi, we would mean ηi jd xid xj≡
d x12
+ d x22
+ d x32
, (2.11)
becauseiandjcan run from 1 ton.
2.2 Minkowski Spacetime 33
||gμν|| =
⎛
⎜⎜
⎝
−1 0 0 0
0 1 0 0
0 0r2 0 0 0 0 r2sin2θ
⎞
⎟⎟
⎠. (2.12)
If we consider a transformation of spatial coordinates only, such as from Cartesian to spherical coordinates or from spherical to cylindrical coordinates, we haveg0i =0.
However, in some cases it may be useful to mix temporal and spatial coordinates and g0i may be non-vanishing.
The results reviewed in Sect.1.3 still hold, but space indices are replaced by spacetime indices. Under the coordinate transformationxμ→xμ, vectors and dual vectors transform as
Vμ→Vμ= ∂xμ
∂xνVν, Vμ→Vμ = ∂xν
∂xμVν. (2.13) Upper indices are now lowered by gμν and lower indices are raised by gμν, for example
Vμ=gμνVν, Vμ=gμνVν. (2.14) The generalization to tensors of any type and order is straightforward.
The trajectory of a point-like particle in the spacetime is a curve called theworld line. Particles moving at the speed of light follow trajectories withds2 =0 by defi- nition, while for particles moving at lower (higher) velocities the line element along their trajectories is ds2<0 (ds2>0). Curves with ds2<0 are called time-like, those withds2=0 are calledlight-likeornull, and those withds2>0 are called space-like.4According to the Einstein Principle of Relativity, there are no particles following space-like trajectories, but in some contexts it is necessary to consider space-like curves.
Figure2.1shows a diagram of the Minkowski spacetime. For simplicity and graph- ical reasons, we consider a spacetime in 1+1 dimensions, so we have the coordinates t andx only. Here the line element isds2= −c2dt2+d x2.P is an event, namely a point in the spacetime. Without loss of generality, we can put P at the origin of the coordinate system. The set of points in the spacetime that are connected toPby a light-like trajectory is called thelight-cone. If we consider a (2+1)-dimensional spacetime(ct,x,y), the light-cone is indeed the surface of a double cone. In 1+ 1 dimensions, like in Fig.2.1, the light-cone is defined by ct= ±x and we have two straight lines. In the case of an (n+1)-dimensional spacetime withn≥3, the light-cone is a hyper-surface. We can also distinguish thefuture light-cone(the set of points of the light-cone witht>0) and thepast light-cone(the set of points of the light-cone witht <0).
4With the conventionημν=diag(1,−1,−1,−1)common in particle physics, the line element along particle trajectories isds2>0 (ds2<0) if the particle moves at a speed lower (higher) than c. In that context, time-like curves haveds2>0 and space-like curves haveds2<0.
Fig. 2.1 Diagram of the Minkowski spacetime. EventPis at the origin of the coordinate system (ct,x). The future light-cone consists of the two half straight linesct= ±xwitht>0. EventBis on the future light-cone and is connected toPby a light-like trajectory. EventAis inside the future light-cone: it is connected toPby a time-like trajectory. Events inside the future light-cone are causally connected to pointP, i.e. they can be affected by what happens inP. EventCis outside the future light-cone: it is connected toPby a space-like trajectory. Events outside the future light-cone are causally disconnected to pointP, i.e. they cannot be affected by what happens inP. The past light-cone consists of the two half straight linesct= ±xwitht<0. EventDis connected toPby a time-like trajectory, eventEis connected toPby a light-like trajectory, and eventFis connected toPby a space-like trajectory.DandEcan affectP, whileFcannot. See the text for more details
According to the Einstein Principle of Relativity, there are no interactions prop- agating faster than light in vacuum. The light cone thus determines the causally connected and disconnected regions with respect to a certain point. With reference to Fig.2.1, an event inPcan influence an event in Aand inB, but cannot influence an event inC. An event in Dor inEcan influence an event in P, while one in F cannot.