(Laplace - Essai philosophique sur les probabilites, Introduction. 1814)

Lorentz Transformations : ,,,,,,,, ,,,,,,,,

The ACTION is a scalar which is the size of an ENERGY for a Time (also infinitesimal interval of time, and returns real numbers ) if the action is local , it must be defined by an Integral.

as the relativistic action of a free material point is proportional to its own time so the relativistic action for a string is proportional to the area of the WorldSheet ie the solutions of the classical equations for the action of a free string are the universe surfaces with minimal area

string tension

metric tensor of target manifold

worldsheet metric

inverse of worldsheet metric

determinant of worldsheet metric

spacelike worldsheet coordinate, its direction is - (negative) by metric signature convention

timelike worldsheet coordinate, its direction is + (positive) by metric signature convention

infinitesimal variation of timelike

the relativistic actione for a

point/particle = zero-dimensional

string = uni-dimensional, featured by a worldsheet

point on space-time

parameters of a point on space-time ( , )

String as vector on space-time

the functions construct the shape of the worldsheet

now, the metric tensor (on (d+1)dimensional space-time) works and we get

= metric tensor on worldsheet

worldsheet area =

if and

then, using the notations

the worldsheet metric tensor is and

the NAMBU-GOTO 's Action analyzes the behaviour of a String by lagrangian mechanics.