String Theory and M-Theory

20 The bosonic string
is time-like, and σ i , which are p space-like coordinates. Since dµp has units
of (length) p+1 the dimension of the p-brane tension is
or energy per unit p-volume.
EXERCISES
[Tp] = (length) −p−1 = mass
(length) p , (2.9)
EXERCISE 2.1
Show that the nonrelativistic limit of the action (2.1) in flat Minkowski
space-time determines the value of the constant α to be the mass of the
point particle.
SOLUTION
In the nonrelativistic limit the action (2.1) becomes
dt
S0 = −α 2 − dx 2 = −α
dt 1 − v 2
≈ −α
dt 1 − 1
2 v2
+ . . . .
Comparing the above expansion with the action of a nonrelativistic point
X
2
X 0
Fig. 2.2. The classical trajectory of a string minimizes the area of the world sheet.
X
1