MTLN[mu,nu,n,nb] denotes the positive component in the
lightcone decomposition of the metric tensor g^{\mu \nu} along the vectors n
and nb. It corresponds to \frac{1}{2} n^{\mu} \bar{n}^\nu.
If one omits n and nb, the program will use
default vectors specified via $FCDefaultLightconeVectorN
and $FCDefaultLightconeVectorNB.
Overview, Pair, FVLP, FVLN, FVLR, SPLP, SPLN, SPLR, MTLP, MTLR.
MTLN[\[Mu], \[Nu], n, nb]\frac{1}{2} \overline{n}^{\mu } \overline{\text{nb}}^{\nu }
StandardForm[MTLN[\[Mu], \[Nu], n, nb] // FCI]\frac{1}{2} \;\text{Pair}[\text{LorentzIndex}[\mu ],\text{Momentum}[n]] \;\text{Pair}[\text{LorentzIndex}[\nu ],\text{Momentum}[\text{nb}]]
Notice that the properties of n and nb
vectors have to be set by hand before doing the actual computation
MTLN[\[Mu], \[Nu], n, nb] FV[p, \[Mu]] // Contract\frac{1}{2} \overline{\text{nb}}^{\nu } \left(\overline{n}\cdot \overline{p}\right)
MTLN[\[Mu], \[Nu], n, nb] FV[p, \[Nu]] // Contract\frac{1}{2} \overline{n}^{\mu } \left(\overline{\text{nb}}\cdot \overline{p}\right)
MTLN[\[Mu], \[Nu], n, nb] FV[n, \[Nu]] // Contract\frac{1}{2} \overline{n}^{\mu } \left(\overline{n}\cdot \overline{\text{nb}}\right)
FCClearScalarProducts[]
SP[n] = 0;
SP[nb] = 0;
SP[n, nb] = 2;MTLN[\[Mu], \[Nu], n, nb] FV[n, \[Nu]] // Contract\overline{n}^{\mu }
FCClearScalarProducts[]