Towards Curved Spacetime - Cosmology Notes

Equivalence Principle¶

1. Weak Equivalence Principle (WEP)¶

Experimental observation shows that inertial mass $m_i$ equals gravitational mass $m_g$ .

For a point particle in an electric potential $\phi_E$ :

m_i \, \vec{a} = -q_e \, \nabla \phi_E

For a point particle in a gravitational potential $\phi_g$ :

m_i \, \vec{a} = -m_g \, \nabla \phi_g

Observation: $m_i = m_g \implies \vec{a} = -\nabla \phi_g$

Consequence: The acceleration due to gravity is universal, independent of the particle’s mass or composition.

2. Free-Falling Frame (FFF)¶

In a region with a constant gravitational field, one can define a frame where gravity disappears locally.
For non-uniform gravity, a local FFF can still be defined.
Gravity manifests as a change of coordinates, i.e., the metric encodes the gravitational field.

3. Einstein Equivalence Principle (EEP)¶

Locally, physics in a free-falling frame is identical to physics in Minkowski spacetime (SR applies).
At any point in spacetime, one can choose coordinates such that the laws of physics take their SR form.

4. Strong Equivalence Principle (SEP)¶

Extends EEP to all laws of nature, including gravitational interactions themselves.

Summary:

Principle	Scope
WEP	Test bodies
EEP	Non-gravitational interactions
SEP	All laws including gravity

Non-Inertial Frames¶

a. Lorentz Transformations¶

In flat spacetime (SR), Lorentz transformations are linear:

x^\mu \to y^\mu = \Lambda^\mu{}_\nu x^\nu

The spacetime interval is invariant:

ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu

In a new frame $y^\mu$ :

dx^\mu = \frac{\partial x^\mu}{\partial y^\alpha} dy^\alpha

Substituting:

ds^2 = \eta_{\mu\nu} \frac{\partial x^\mu}{\partial y^\alpha} \frac{\partial x^\nu}{\partial y^\beta} dy^\alpha dy^\beta

For linear Lorentz transformations, this reduces to:

\eta_{\mu\nu} \frac{\partial x^\mu}{\partial y^\alpha} \frac{\partial x^\nu}{\partial y^\beta} = \eta_{\alpha\beta}

b. General Curvilinear Transformations¶

For a general invertible transformation:

x^\mu \to y^\mu(x^\alpha)

Jacobian matrix:

J^\mu{}_\nu = \frac{\partial y^\mu}{\partial x^\nu}

The metric in the new frame $R'$ is:

g_{\alpha\beta}(y) = \eta_{\mu\nu} \frac{\partial x^\mu}{\partial y^\alpha} \frac{\partial x^\nu}{\partial y^\beta}

If $R'$ is a FFF, the metric encodes the gravitational interaction.

c. Motion of a Particle in Non-Inertial Frame¶

In an inertial frame $R$ (or FFF in gravity):

a^\mu = \frac{d^2 x^\mu}{d\tau^2} = 0

In a general frame $y^\mu$ :

\frac{d^2 y^\alpha}{d\tau^2} = \frac{\partial y^\alpha}{\partial x^\mu} \frac{d^2 x^\mu}{d\tau^2} + \frac{\partial^2 y^\alpha}{\partial x^\mu \partial x^\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau}

Since $d^2 x^\mu / d\tau^2 = 0$ , we get:

\frac{d^2 y^\alpha}{d\tau^2} + \underbrace{\frac{\partial^2 y^\alpha}{\partial x^\mu \partial x^\nu} \frac{\partial x^\mu}{\partial y^\beta} \frac{\partial x^\nu}{\partial y^\gamma}}_{\Gamma^\alpha_{\beta\gamma}} \frac{dy^\beta}{d\tau} \frac{dy^\gamma}{d\tau} = 0

This defines the geodesic equation in the new frame:

\frac{d^2 y^\alpha}{d\tau^2} + \Gamma^\alpha_{\beta\gamma} \frac{dy^\beta}{d\tau} \frac{dy^\gamma}{d\tau} = 0

where $\Gamma^\alpha_{\beta\gamma}$ are the Christoffel symbols (affine connection).

The parameter $\tau$ is the affine parameter, usually the proper time for massive particles.

Summary¶

Gravity can be interpreted as curvature of spacetime.
Free-fall corresponds to geodesic motion.
Christoffel symbols encode the effects of non-inertial (curved) frames.
This is the geometric foundation of General Relativity.