Gravity and Strings

50 A perturbative introduction to general relativity
Indeed, P µ Pµ is a constant: using the definition of momentum, we find, without using the
equations of motion, the mass-shell condition
P µ Pµ = M 2 c 2 , (3.20)
and we have just shown that this constraint can be understood as a consequence of reparametrization
invariance.
There are two special parameters one can use. 7 One is the particle’s proper time (or
length) ξ = s, defined by the property
ηµν ˙X µ ˙X ν = 1. (3.21)
Owing to this definition, the action is usually written as
Spp[X (s)] =−Mc ds. (3.22)
Although this form is unsuitable for finding the equations of motion, it tells us that the
action of a massive point-particle is proportional to its worldline’s proper length, and the
minimal-action principle tells us that the particle moves along worldlines of minimal proper
length. Observe that, from the quantum mechanics point of view, since the measure in the
path integral is the exponential of
i Mc
S = i
ds =
i
−λCompton
ds, (3.23)
the proper length is measured in units of the particle’s reduced Compton wavelength.
The second special parameter that we can use is the coordinate time ξ = X 0 = cT. This
choice of gauge fixes one of the particle’s coordinates X 0 (ξ) = ξ.Inthis gauge (The physical
or static gauge) one can study the non-relativistic limit ˙X i ˙X i = (v/c) 2 ≪ 1. In this limit
the action (3.8) becomes, up to a total derivative, the non-relativistic action of a particle:
S[X i
(t)] = dt 1
2 Mv2 − Mc2 . (3.24)
3.1.3 The massive point-particle coupled to scalar gravity
The coupling to the scalar gravitational field is dictated by the action Eq. (3.3). We compute
the energy–momentum tensor using Rosenfeld’s prescription (Section 2.4.3):
T µν
˙X
pp (x) =−Mc2 dξ
µ ˙X ν
δ
˙X ρ ˙X σ
(d) [X (ξ) − x], (3.25)
ηρσ
which is conserved, as one can prove by using the equations of motion. The trace is identical
to the Lagrangian, 8 and thus the action for the coupled particle-plus-gravity system
7 Purists call the same curve with two different parametrizations different curves, but from a physical point of
view they are clearly the same object.
8 Observe that, in the static gauge, the 00 component of this tensor gives Eq. (3.6).