A.4 More on in­dex no­ta­tion

En­gi­neer­ing stu­dents are of­ten much more fa­mil­iar with lin­ear al­ge­bra than with ten­sor al­ge­bra. So it may be worth­while to look at the Lorentz trans­for­ma­tion from a lin­ear al­ge­bra point of view. The re­la­tion to ten­sor al­ge­bra will be in­di­cated. If you do not know lin­ear al­ge­bra, there is lit­tle point in read­ing this ad­den­dum.

A con­travari­ant four-vec­tor like po­si­tion can be pic­tured as a col­umn vec­tor that trans­forms with the Lorentz ma­trix $\Lambda$. A co­vari­ant four-vec­tor like the gra­di­ent of a scalar func­tion can be pic­tured as a row vec­tor that trans­forms with the in­verse Lorentz ma­trix $\Lambda^{-1}$:

$$\kern-1pt{\buildrel\raisebox{-1.5pt}[0pt][0pt] {\hbox{\hspa... ...over\nabla} \kern-1.3ptf\Big)_{\rm {A}}^{\rm {T}} \Lambda^{-1}$$

In lin­ear al­ge­bra, a su­per­script $T$ trans­forms columns into rows and vice-versa. Since you think of the gra­di­ent by it­self as a col­umn vec­tor, the T turns it into a row vec­tor. Note also that putting the fac­tors in a prod­uct in the cor­rect or­der is es­sen­tial in lin­ear al­ge­bra. In the sec­ond equa­tion above, the gra­di­ent, writ­ten as a row, pre­mul­ti­plies the in­verse Lorentz ma­trix.

In ten­sor no­ta­tion, the above ex­pres­sions are writ­ten as

$$x^\mu_{\rm {B}} = \lambda{}^\mu{}_\nu x^\nu_{\rm {A}} \qqu... ...\partial_{\nu,\rm {A}} f \left(\lambda^{-1}\right){}^\nu{}_\mu$$

The or­der of the fac­tors is now no longer a con­cern; the cor­rect way of mul­ti­ply­ing fol­lows from the names of the in­dices.

The key prop­erty of the Lorentz trans­for­ma­tion is that it pre­serves dot prod­ucts. Pretty much every­thing else fol­lows from that. There­fore the dot prod­uct must now be for­mu­lated in terms of lin­ear al­ge­bra. That can be done as fol­lows:

$$\kern-1pt{\buildrel\raisebox{-1.5pt}[0pt][0pt] {\hbox{\hspa... ...& 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$

The ma­trix $G$ is called the “Minkowski met­ric.” The ef­fect of $G$ on $\kern-1pt{\buildrel\raisebox{-1.5pt}[0pt][0pt] {\hbox{\hspace{1pt}$\scriptscriptstyle\hookrightarrow$\hspace{0pt}}}\over r} \kern-1.3pt_2$ is to flip over the sign of the ze­roth, time, en­try. Look­ing at it an­other way, the ef­fect of $G$ on the pre­ced­ing $\kern-1pt{\buildrel\raisebox{-1.5pt}[0pt][0pt] {\hbox{\hspace{1pt}$\scriptscriptstyle\hookrightarrow$\hspace{0pt}}}\over r} \kern-1.3pt_1^{ \rm {T}}$ is to flip over the sign of its ze­roth en­try. Ei­ther way, $G$ pro­vides the mi­nus sign for the prod­uct of the time co­or­di­nates in the dot prod­uct.

In ten­sor no­ta­tion, the above ex­pres­sion must be writ­ten as

$$\kern-1pt{\buildrel\raisebox{-1.5pt}[0pt][0pt] {\hbox{\hspa... ...0pt}}}\over r} \kern-1.3pt_2 \equiv x_1^\mu g_{\mu\nu} x_2^\nu$$

In par­tic­u­lar, since space-time po­si­tions have su­per­scripts, the met­ric ma­trix $G$ needs to be as­signed sub­scripts. That main­tains the con­ven­tion that a sum­ma­tion in­dex ap­pears once as a sub­script and once as a su­per­script.

Since dot prod­ucts are in­vari­ant,

$$\kern-1pt{\buildrel\raisebox{-1.5pt}[0pt][0pt] {\hbox{\hspa... ...e\hookrightarrow$\hspace{0pt}}}\over r} \kern-1.3pt_{2\rm {A}}$$

Here the fi­nal equal­ity sub­sti­tuted the Lorentz trans­for­ma­tion from A to B. Re­call that if you take a trans­pose of a prod­uct, the or­der of the fac­tors gets in­verted. If the ex­pres­sion to the far left is al­ways equal to the one to the far right, it fol­lows that

$$\fbox{$\displaystyle \Lambda^{\rm{T}} G \Lambda = G $} %$$ (A.13)

This must be true for any Lorentz trans­form. In fact, many sources de­fine Lorentz trans­forms as trans­forms that sat­isfy the above re­la­tion­ship. There­fore, this re­la­tion­ship will be called the defin­ing re­la­tion. It is very con­ve­nient for do­ing the var­i­ous math­e­mat­ics. How­ever, this sort of ab­stract de­f­i­n­i­tion does not re­ally pro­mote easy phys­i­cal un­der­stand­ing.

And there are a cou­ple of other prob­lems with the defin­ing re­la­tion. For one, it al­lows Lorentz trans­forms in which one ob­server uses a left-handed co­or­di­nate sys­tem in­stead of a right-handed one. Such an ob­server ob­serves a mir­ror im­age of the uni­verse. Math­e­mat­i­cally at least. A Lorentz trans­form that switches from a nor­mal right-handed co­or­di­nate sys­tem to a left handed one, (or vice-versa), is called “im­proper.” The sim­plest ex­am­ple of such an im­proper trans­for­ma­tion is $\Lambda$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$$G$. That is called the “par­ity trans­for­ma­tion.” Its ef­fect is to flip over all spa­tial po­si­tion vec­tors. (If you make a pic­ture of it, you can see that in­vert­ing the di­rec­tions of the $x$, $y$, and $z$ axes of a right-handed co­or­di­nate sys­tem pro­duces a left-handed sys­tem.) To see that $\Lambda$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$$G$ sat­is­fies the defin­ing re­la­tion above, note that $G$ is sym­met­ric, $G^{\rm {T}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $G$, and its own in­verse, $GG$ $\vphantom0\raisebox{1.5pt}{$=$}$ $I$.

An­other prob­lem with the defin­ing re­la­tion is that it al­lows one ob­server to use an in­verted di­rec­tion of time. Such an ob­server ob­serves the uni­verse evolv­ing to smaller val­ues of her time co­or­di­nate. A Lorentz trans­form that switches the di­rec­tion of time from one ob­server to the next is called “nonorthochro­nous.” (Or­tho in­di­cates cor­rect, and chro­nous time.) The sim­plest ex­am­ple of a nonorthochro­nous trans­for­ma­tion is $\Lambda$ $\vphantom0\raisebox{1.5pt}{$=$}$ $G$. That trans­for­ma­tion is called “time-re­ver­sal.” Its ef­fect is to sim­ply re­place the time $t$ by $\vphantom{0}\raisebox{1.5pt}{$-$}$$t$. It sat­is­fies the defin­ing re­la­tion for the same rea­sons as the par­ity trans­for­ma­tion.

As a re­sult, there are four types of Lorentz trans­for­ma­tions that sat­isfy the defin­ing re­la­tion. First of all there are the nor­mal proper or­thochro­nous ones. The sim­plest ex­am­ple is the unit ma­trix $I$, cor­re­spond­ing to the case that the ob­servers A and B are iden­ti­cal. Sec­ond, there are the im­proper ones like $\vphantom{0}\raisebox{1.5pt}{$-$}$$G$ that switch the hand­ed­ness of the co­or­di­nate sys­tem. Third there are the nonorthochro­nous ones like $G$ that switch the cor­rect di­rec­tion of time. And fourth, there are im­proper nonorthochro­nous trans­forms, like $\vphantom{0}\raisebox{1.5pt}{$-$}$$GG$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$$I$, that switch both the hand­ed­ness and the di­rec­tion of time.

These four types of Lorentz trans­forms form four dis­tinct groups. You can­not grad­u­ally change from a right-handed co­or­di­nate sys­tem to a left-handed one. Ei­ther a co­or­di­nate sys­tem is right-handed or it is left-handed. There is noth­ing in be­tween. By the same to­ken, ei­ther a co­or­di­nate sys­tem has the proper di­rec­tion of time or the ex­actly op­po­site di­rec­tion.

These four groups are re­flected in math­e­mat­i­cal prop­er­ties of the Lorentz trans­forms. Lorentz trans­form ma­tri­ces have de­ter­mi­nants that are ei­ther 1 or $\vphantom{0}\raisebox{1.5pt}{$-$}$1. That is eas­ily seen from tak­ing de­ter­mi­nants of both sides of the defin­ing equa­tion (A.13), split­ting the left de­ter­mi­nant in its three sep­a­rate fac­tors. Also, Lorentz trans­forms have val­ues of the en­try $\lambda{}^0{}_0$ that are ei­ther greater or equal to 1 or less or equal to $\vphantom{0}\raisebox{1.5pt}{$-$}$1. That is read­ily seen from writ­ing out the ${}^0{}_0$ en­try of (A.13).

Proper or­thochro­nous Lorentz trans­forms have a de­ter­mi­nant 1 and an en­try $\lambda{}^0{}_0$ greater or equal to 1. That can read­ily be checked for the sim­plest ex­am­ple $\Lambda$ $\vphantom0\raisebox{1.5pt}{$=$}$ $I$. More gen­er­ally, it can eas­ily be checked that $\lambda{}^0{}_0$ is the time di­lata­tion fac­tor for events that hap­pen right in the hands of ob­server A. That is the phys­i­cal rea­son that $\lambda{}^0{}_0$ must al­ways be greater or equal to 1. Trans­forms that have $\lambda{}^0{}_0$ less or equal to $\vphantom{0}\raisebox{1.5pt}{$-$}$1 flip over the cor­rect di­rec­tion of time. So they are nonorthochro­nous. Trans­forms that switch over the hand­ed­ness of the co­or­di­nate sys­tem pro­duce a neg­a­tive de­ter­mi­nant. But so do nonorthochro­nous trans­forms. If a trans­form flips over both hand­ed­ness and the di­rec­tion of time, it has a time di­lata­tion less or equal to $\vphantom{0}\raisebox{1.5pt}{$-$}$1 but a pos­i­tive de­ter­mi­nant.

For rea­sons given above, if you start with some proper or­thochro­nous Lorentz trans­form like $\Lambda$ $\vphantom0\raisebox{1.5pt}{$=$}$ $I$ and grad­u­ally change it, it stays proper and or­thochro­nous. But in ad­di­tion its de­ter­mi­nant stays 1 and its time-di­lata­tion en­try stays greater or equal to 1. The rea­sons are es­sen­tially the same as be­fore. You can­not grad­u­ally change from a value of 1 or above to a value of $\vphantom{0}\raisebox{1.5pt}{$-$}$1 or be­low if there is noth­ing in be­tween.

One con­se­quence of the defin­ing re­la­tion (A.13) mer­its men­tion­ing. If you pre­mul­ti­ply both sides of the re­la­tion by $G^{-1}$, you im­me­di­ately see that

$$\fbox{$\displaystyle \Lambda^{-1} = G^{-1} \Lambda^T G $}$$ (A.14)

This is the easy way to find in­verses of Lorentz trans­forms. Also, since $G^2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $I$, $G^{-1}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $G$. How­ever, it can­not hurt to leave the ex­pres­sion as writ­ten. There are other ap­pli­ca­tions in ten­sor al­ge­bra in which $G^{-1}$ is not equal to $G$.

As al­ready il­lus­trated above, what mul­ti­pli­ca­tions by $G$ do is flip over the sign of some en­tries. So to find an in­verse of a Lorentz trans­form, just flip over the right en­tries. To be pre­cise, flip over the en­tries in which one in­dex is 0 and the other is not.

The above ob­ser­va­tions can be read­ily con­verted to ten­sor no­ta­tion. First an equiv­a­lent is needed to some de­f­i­n­i­tions used in ten­sor al­ge­bra but not nor­mally in lin­ear al­ge­bra. The “ low­ered cov­ec­tor” to a con­travari­ant vec­tor like po­si­tion will be de­fined as

$$\kern-1pt{\buildrel\raisebox{-1.5pt}[0pt][0pt] {\hbox{\hspa... ...ookrightarrow$\hspace{0pt}}}\over r} \kern-1.3pt^{ \rm {T}} G$$

In words, take a trans­pose and post­mul­ti­ply with the met­ric $G$. The re­sult is a row vec­tor while the orig­i­nal is a col­umn vec­tor.

Note that the dot prod­uct can now be writ­ten as

$$\kern-1pt{\buildrel\raisebox{-1.5pt}[0pt][0pt] {\hbox{\hspa... ...criptstyle\hookrightarrow$\hspace{0pt}}}\over r} \kern-1.3pt_2$$

Note also that low­ered cov­ec­tors are co­vari­ant vec­tors; they are row vec­tors that trans­form with the in­verse Lorentz trans­form. To check that, sim­ply plug in the Lorentz trans­for­ma­tion of the orig­i­nal vec­tor and use the ex­pres­sion for the in­verse Lorentz trans­form above.

Sim­i­larly, the raised con­travec­tor to a co­vari­ant vec­tor like a gra­di­ent will be de­fined as

$$\Big(\kern-1pt{\buildrel\raisebox{-1.5pt}[0pt][0pt] {\hbox{... ...ptstyle\hookrightarrow$\hspace{0pt}}}\over\nabla} \kern-1.3ptf$$

In words, take a trans­pose and pre­mul­ti­ply by the in­verse met­ric. The raised con­travec­tor is a con­travari­ant vec­tor. Form­ing a raised con­travec­tor of a low­ered cov­ec­tor gives back the orig­i­nal vec­tor. And vice-versa. (Note that met­rics are sym­met­ric ma­tri­ces in check­ing that.)

In ten­sor no­ta­tion, the low­ered cov­ec­tor is writ­ten as

$$x_\mu = x^\nu g_{\nu\mu}$$

Note that the graph­i­cal ef­fect of mul­ti­ply­ing by the met­ric ten­sor is to lower the vec­tor in­dex.

Sim­i­larly, the raised con­travec­tor to a cov­ec­tor is

$$\partial^\mu f = \left(g^{-1}\right)^{\mu\nu} \partial_\nu f$$

It shows that the in­verse met­ric can be used to raise in­dices. But do not for­get the golden rule: rais­ing or low­er­ing an in­dex is much more than cos­metic: you pro­duce a fun­da­men­tally dif­fer­ent vec­tor.

(That is not true for so-called “Carte­sian ten­sors” like purely spa­tial po­si­tion vec­tors. For these the met­ric $G$ is the unit ma­trix. Then rais­ing or low­er­ing an in­dex has no real ef­fect. By the way, the unit ma­trix is in ten­sor no­ta­tion $\delta{}^\mu{}_\nu$. That is called the Kro­necker delta. Its en­tries are 1 if the two in­dices are equal and 0 oth­er­wise.)

Us­ing the above no­ta­tions, the dot prod­uct be­comes as stated in chap­ter 1.2.5,

$$x_{1,\mu} x_2^\mu$$

More in­ter­est­ingly, con­sider the in­verse Lorentz trans­form. Ac­cord­ing to the ex­pres­sion given above $\Lambda^{-1}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $G^{-1}\Lambda^{\rm {T}}G$, so:

$$\left(\lambda^{-1}\right){}^\mu{}_\nu = \left(g^{-1}\right){}^{\mu\alpha} \lambda{}^\beta{}_\alpha g{}_{\beta\nu}$$

(A trans­pose of a ma­trix, in this case $\Lambda$, swaps the in­dices.) Ac­cord­ing the in­dex rais­ing/low­er­ing con­ven­tions above, in the right hand side the heights of the in­dices of $\Lambda$ are in­verted. So you can de­fine a new ma­trix with en­tries

$$\fbox{$\displaystyle \lambda{}_\nu{}^\mu \equiv \left(\lambda^{-1}\right){}^\mu{}_\nu $}$$

But note that the so-de­fined ma­trix is not the Lorentz trans­form ma­trix:

$$\lambda{}_\nu{}^\mu \ne \lambda{}^\mu{}_\nu$$

It is a dif­fer­ent ma­trix. In par­tic­u­lar, the signs on some en­tries are swapped.

(Need­less to say, var­i­ous sup­pos­edly au­thor­i­ta­tive sources list both ma­tri­ces as $\lambda{}_\nu^\mu$ for that ex­quis­ite fi­nal bit of con­fu­sion. It is ap­par­ently not easy to get sub­scripts and su­per­scripts straight if you use some hor­ri­ble prod­uct like MS Word. Of course, the sim­ple an­swer would be to use a place holder in the empty po­si­tion that in­di­cates whether or not the in­dex has been raised or low­ered. For ex­am­ple:

$$\lambda{}_{\nu R}^{L\mu} \ne \lambda{}^{\mu N}_{N\nu}$$

How­ever, this is not pos­si­ble be­cause it would add clar­ity.)

Now con­sider an­other very con­fus­ing re­sult. Start with

$$G^{-1}G G^{-1} = G^{-1} \qquad\Longrightarrow\qquad \left(... ...ft(g^{-1}\right){}^{\beta\nu} = \left(g^{-1}\right){}^{\mu\nu}$$

Ac­cord­ing to the rais­ing con­ven­tions, that can be writ­ten as

$$\fbox{$\displaystyle g{}^{\mu\nu} \equiv \left(g^{-1}\right){}^{\mu\nu} $}$$ (A.15)

Does this not look ex­actly as if $G$ $\vphantom0\raisebox{1.5pt}{$=$}$ $G^{-1}$? That may be true in the case of Lorentz trans­forms and the as­so­ci­ated Minkowski met­ric. But for more gen­eral ap­pli­ca­tions of ten­sor al­ge­bra it is most def­i­nitely not true. Al­ways re­mem­ber the golden rule: names of ten­sors are only mean­ing­ful if the in­dices are at the right height. The right height for the in­dices of $G$ is sub­scripts. So $g^{\mu\nu}$ does not in­di­cate an en­try of $G$. In­stead it turns out to rep­re­sent an en­try of $G^{-1}$.

So physi­cists now have two op­tions. They can write the en­tries of $G^{-1}$ in the un­der­stand­able form $\left(g^{-1}\right){}^{\mu\nu}$. Or they can use the con­fus­ing, er­ror-prone form $g{}^{\mu\nu}$. So what do you think they all do? If you guessed op­tion (b), you are mak­ing real progress in your study of mod­ern physics.

Of­ten the best way to ver­ify some ar­cane ten­sor ex­pres­sion is to con­vert it to lin­ear al­ge­bra. (Re­mem­ber to check the heights of the in­dices when do­ing so. If they are on the wrong height, re­store the omit­ted fac­tor $g_{..}$ or $\left(g^{-1}\right){}^{..}$.) Some ad­di­tional re­sults that are use­ful in this con­text are

$$\Lambda^{-\rm {T}} G \Lambda^{-1} = G \qquad \Lambda G^{-... ...da^{\rm {T}} = G^{-1} \qquad \Lambda G \Lambda^{\rm {T}} = G$$

The first of these im­plies that the in­verse of a Lorentz trans­form is a Lorentz trans­form too. That is read­ily ver­i­fied from the defin­ing re­la­tion (A.13) by pre­mul­ti­ply­ing by $\Lambda^{-\rm {T}}$ and post­mul­ti­ply­ing by $\Lambda^{-1}$. The sec­ond ex­pres­sion is sim­ply the ma­trix in­verse of the first. Both of these ex­pres­sions gen­er­al­ize to any sym­met­ric met­ric $G$. The fi­nal ex­pres­sion im­plies that the trans­pose of a Lorentz trans­form is a Lorentz trans­form too. That is only true for Lorentz trans­forms and the as­so­ci­ated Minkowski met­ric. Or ac­tu­ally, it is also true for any other met­ric in which $G^{-1}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $G$, in­clud­ing Carte­sian ten­sors. For these met­rics, the fi­nal ex­pres­sion above is the same as the sec­ond ex­pres­sion.