Introduction

There are two major conventions for the metric signature: (+,−,−,−)\p{+,-,-,-} (west coast) and (−,+,+,+)\p{-,+,+,+} (east coast). However, the first convention that I have met in my journey of learning physics is neither of them: the imaginary time. Shortly after, I started using the west coast convention, so I never really used the imaginary time convention seriously. I personally dislike the imaginary time convention, and so do most people in the physics community and history, which is why most modern textbooks use either the west coast or the east coast convention. One of my past physics teachers deemed the imaginary time convention to be a heresy (异端邪说).

The teacher’s writing

However, in some cases, the imaginary time convention can be convenient due to the use of multi-index notation (which is more concise and feature-rich than the Einstein notation). Here is one of such cases: the derivation of the metric in Poincaré coordinates for the anti-de Sitter space.

The dd-dimensional anti-de Sitter space AdSd\mrm{AdS}_d of scale ll is defined as the hyperboloid −l2=−T12−T22+∑i=1d−1(Xi)2-l^2=-T_1^2-T_2^2+\sum_{i=1}^{d-1}\p{X^i}^2 in Md−1,2M^{d-1,2} (the analogue of the Minkowski space, but with signature d−1,2d-1,2). The Poincaré coordinates are defined as z≔l2T1+Xd−1,t≔lT2T1+Xd−1,xi≔lXiT1+Xd−1,i=1,…,d−2.\begin{align*} z&\ceq\fr{l^2}{T_1+X^{d-1}},\\ t&\ceq\fr{lT_2}{T_1+X^{d-1}},\\ x^i&\ceq\fr{lX^i}{T_1+X^{d-1}},&i=1,\ldots,d-2. \end{align*}

The derivation

Define T≔T1T\ceq T_1 and X≔Xd−1X\ceq X^{d-1} just for fun. Then, define two (d−1)\p{d-1}-dimensional multi-indices Y≔(iT2,X1,…,Xd−2),y≔(it,x1,…,xd−2).Y\ceq\p{\i T_2,X^1,\ldots,X^{d-2}},\quad y\ceq\p{\i t,x^1,\ldots,x^{d-2}}.

The hyperboloid constraint and the metric (east coast convention) are then X2−T2+Y2=−l2,ds2=dX2−dT2+dY2,X^2-T^2+Y^2=-l^2,\quad \d s^2=\d X^2-\d T^2+\d Y^2, which are equivalently (X+T)(X−T)=−l2−Y2,ds2=(dX+dT)(dX−dT)+dY2.\p{X+T}\p{X-T}=-l^2-Y^2,\quad\d s^2=\p{\d X+\d T}\p{\d X-\d T}+\d Y^2. (1)(1) The definition of the Poincaré coordinates can be written as z=l2X+T,y=zlY,z=\fr{l^2}{X+T},\quad y=\fr zlY, or equivalently X+T=l2z,Y=lyz.X+T=\fr{l^2}z,\quad Y=\fr{ly}z. (2)(2)

Substitute Equation 2 into the first equation in Equation 1. Then, we have X−T=−z−y2z.X-T=-z-\fr{y^2}z. (3)(3) Differentiate Equation 2 and 3, and we have dX+dT=−l2z2 dz,dX−dT=−dz+y2z2 dz−2yz dy,dY=l(dyz−yz2 dz).\d X+\d T=-\fr{l^2}{z^2}\,\d z,\quad \d X-\d T=-\d z+\fr{y^2}{z^2}\,\d z-\fr{2y}z\,\d y,\quad \d Y=l\p{\fr{\d y}z-\fr y{z^2}\,\d z}. Substitute this into the second equation in Equation 1, and we have ds2=−l2z2dz(−dz+y2z2 dz−2yz dy)+l2(dyz−yz2 dz)2=l2z2(dy2+dz2).\d s^2=-\fr{l^2}{z^2}\d z\p{-\d z+\fr{y^2}{z^2}\,\d z-\fr{2y}z\,\d y}+l^2\p{\fr{\d y}z-\fr y{z^2}\,\d z}^2 =\fr{l^2}{z^2}\p{\d y^2+\d z^2}.

Finally, substitute back the definition of yy, and we have the result ds2=l2z2(−dt2+∑i=1d−2(dxi)2+dz2).\d s^2=\fr{l^2}{z^2}\p{-\d t^2+\sum_{i=1}^{d-2}\p{\d x^i}^2+\d z^2}.