# #3 Nambu-Goto action

Given a -dimensional manifold with a metric , a string is a one-dimensional object moving inside . The Nambu-Goto action describes the dynamics of these objects. It’s strongly inspired in the relativistic dynamics of a point particle.

A point particle describes a curve when it’s moving through space and time. Such curve (its trajectory) depends on a single parameter, usually the proper time. Likewise, a string describes a two-dimensional surface called *world sheet*, and depends on two parameters . Even though the role of both parameters is arbitrary, one can think of as the proper time, and, for any given , would label each one of the points that the string is made of.

# #2 – Hilbert – Einstein action – Part 2

Now we compute . First, the Riemann tensor has the following definition:

so

# Self-portrait

# #2 – Hilbert – Einstein action – Part 1

The Hilbert-Einstein action is:

Einstein equations (the evolution equations of the metrig ) can be found if we set .

# #1 – Proof: variation of a determinant

By definition:

so

Thus:

If is a metric tensor, so we have:

Equating terms proportional to :

Finally: