Non-Euclidean Geometry and General Relativity

Part I: Manifolds and Coordinate Systems¶

1.1 Curved Spacetime as a Manifold¶

In General Relativity, spacetime is modeled as a manifold $\mathcal{M}$ .

Key Properties:

$\mathcal{M}$ is not a vector space
There is no global Cartesian coordinate system covering the entire manifold

Important Insight: At each point $p \in \mathcal{M}$ , we can define a tangent space $T_p\mathcal{M}$ that is a vector space.

1.2 General Coordinate Systems¶

The manifold $\mathcal{M}$ is covered by arbitrary coordinates:

x^\mu = (x^0, x^1, x^2, x^3)

(1)

Crucial Note: These coordinates have no intrinsic physical meaning; only tensorial quantities constructed from them are physically meaningful.

Part II: Tangent and Cotangent Spaces¶

2.1 Tangent Space $T_p\mathcal{M}$ ¶

At each point $p \in \mathcal{M}$ , the tangent space $T_p\mathcal{M}$ is defined as:

A vector space with dimension equal to $\dim(\mathcal{M})$
Spanned by the coordinate derivatives

Natural Basis:

e_\mu \equiv \frac{\partial}{\partial x^\mu} \equiv \partial_\mu

(2)

Vector Expansion: Any vector $v \in T_p\mathcal{M}$ can be expanded as:

v = v^\mu e_\mu = v^\mu \partial_\mu

(3)

The components $v^\mu$ are called contravariant components.

2.2 Coordinate Transformations for Vectors¶

Consider a coordinate transformation:

x^\mu \to x'^\mu(x)

(4)

Basis Transformation:

\partial_\mu = \frac{\partial x'^\nu}{\partial x^\mu} \partial'_\nu

(5)

Component Transformation (to keep $v$ invariant):

v'^\mu = \frac{\partial x'^\mu}{\partial x^\nu} v^\nu

(6)

2.3 Cotangent Space $T_p^*\mathcal{M}$ ¶

The cotangent space $T_p^*\mathcal{M}$ is the dual space of the tangent space.

Basis One-Forms: $dx^\mu$

Duality Pairing:

dx^\mu(\partial_\nu) = \delta^\mu_\nu

(7)

Covector Expansion: Any covector (1-form) $\lambda \in T_p^*\mathcal{M}$ can be written as:

\lambda = \lambda_\mu dx^\mu

(8)

The components $\lambda_\mu$ are called covariant components.

2.4 Transformation of Covectors¶

Since the scalar product $\lambda(v)$ must be invariant:

\lambda'_\mu = \frac{\partial x^\nu}{\partial x'^\mu} \lambda_\nu

(9)

Note the inverse Jacobian, opposite to vector transformation.

Part III: Tensors and the Metric¶

3.1 General Tensors¶

A tensor of type $(p,q)$ has components:

T^{\mu_1 \dots \mu_p}{}_{\nu_1 \dots \nu_q}

(10)

Examples:

Scalar: type $(0,0)$
Vector: type $(1,0)$
Covector: type $(0,1)$
Metric: type $(0,2)$

Transformation Rule:

T'^{\mu_1 \dots \mu_p}{}_{\nu_1 \dots \nu_q} = \frac{\partial x'^{\mu_1}}{\partial x^{\alpha_1}} \dots \frac{\partial x'^{\mu_p}}{\partial x^{\alpha_p}} \frac{\partial x^{\beta_1}}{\partial x'^{\nu_1}} \dots \frac{\partial x^{\beta_q}}{\partial x'^{\nu_q}} T^{\alpha_1 \dots \alpha_p}{}_{\beta_1 \dots \beta_q}

(11)

3.2 Tensor Contraction¶

Partial Contraction (yields new tensor):

U^\mu{}_\rho = T^{\mu\nu} S_{\nu\rho}

(12)

Full Contraction (yields scalar):

\phi = T^{\mu\nu} S_{\mu\nu}

(13)

3.3 Metric Tensor and Distances¶

The metric tensor $g_{\mu\nu}$ is a symmetric $(0,2)$ tensor:

g = g_{\mu\nu} dx^\mu \otimes dx^\nu

(14)

Spacetime Interval:

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

(15)

3.4 Raising and Lowering Indices¶

Lowering Indices:

v_\mu = g_{\mu\nu} v^\nu

(16)

Raising Indices:

v^\mu = g^{\mu\nu} v_\nu

(17)

where $g^{\mu\nu}$ is the inverse metric:

g^{\mu\alpha} g_{\alpha\nu} = \delta^\mu_\nu

(18)

3.5 Determinant of the Metric¶

Under coordinate transformation:

g'_{\mu\nu} = \frac{\partial x^\alpha}{\partial x'^\mu} \frac{\partial x^\beta}{\partial x'^\nu} g_{\alpha\beta}

(19)

Determinant Transformation:

\det g' = (\det J)^{-2} \det g

(20)

where $J$ is the Jacobian matrix $\partial x'^\mu/\partial x^\nu$ .

Important: $\det g$ is not a scalar.

Part IV: Covariance and Local Frames¶

4.1 Covariance of Physical Laws¶

Physical laws must be expressed as tensor equations to ensure coordinate independence.

If a tensor equation holds in one coordinate system:

\xi^\rho{}_{\mu\nu} = 0

(21)

then in any other coordinates:

\xi'^\rho{}_{\mu\nu} = 0

(22)

4.2 Local Free-Falling Frames (LFFF)¶

At any point $p \in \mathcal{M}$ , we can choose coordinates (Riemann normal coordinates) such that:

g_{\mu\nu}(p) = \eta_{\mu\nu}, \quad \partial_\alpha g_{\mu\nu}(p) = 0

(23)

Implications:

Locally, physics reduces to Special Relativity
Globally, curvature cannot be removed

Part V: Covariant Derivative¶

5.1 The Problem with Partial Derivatives¶

For Scalars (works fine):

\partial_\mu f \quad \text{is a tensor}

(24)

For Vectors (problematic):

\partial_\alpha v^\mu \quad \text{is NOT a tensor}

(25)

Extra terms appear due to derivatives of the Jacobian.

5.2 Defining the Covariant Derivative¶

For a Vector $v^\mu$ :

\nabla_\alpha v^\mu = \partial_\alpha v^\mu + \Gamma^\mu_{\alpha\beta} v^\beta

(26)

The $\Gamma^\mu_{\alpha\beta}$ are connection coefficients.

For a Covector $\lambda_\mu$ :

\nabla_\alpha \lambda_\mu = \partial_\alpha \lambda_\mu - \Gamma^\beta_{\alpha\mu} \lambda_\beta

(27)

5.3 Properties of Covariant Derivative¶

Linearity: For tensors $T$ , $S$ and scalars $a$ , $b$ :
$\nabla_\mu (a\,T + b\,S) = a\,\nabla_\mu T + b\,\nabla_\mu S$
(28)
Leibniz Rule: For tensors $T$ and $S$ :
$\nabla_\mu (T \otimes S) = (\nabla_\mu T) \otimes S + T \otimes (\nabla_\mu S)$
(29)
Action on Scalars: For scalar field $f$ :
$\nabla_\mu f = \partial_\mu f$
(30)

These properties uniquely determine $\nabla$ on all tensors.

Part VI: Levi-Civita Connection¶

6.1 Determining the Connection¶

The connection is fixed by imposing:

Symmetry (Torsion-Free):
$\Gamma^\alpha_{\mu\nu} = \Gamma^\alpha_{\nu\mu}$
(31)
Metric Compatibility:
$\nabla_\alpha g_{\mu\nu} = 0$
(32)

6.2 Christoffel Symbols¶

These conditions yield the unique Levi-Civita connection:

\Gamma^\alpha_{\mu\nu} = \frac{1}{2} g^{\alpha\beta} \left( \partial_\mu g_{\nu\beta} + \partial_\nu g_{\mu\beta} - \partial_\beta g_{\mu\nu} \right)

(33)

Important: $\Gamma^\alpha_{\mu\nu}$ is not a tensor.

6.3 Useful Identities¶

Divergence of a Vector:

\nabla_\mu v^\mu = \frac{1}{\sqrt{-g}} \partial_\mu\!\left(\sqrt{-g}\, v^\mu\right)

(34)

Scalar Wave Operator (d’Alembertian):

\square \phi = \nabla_\mu \nabla^\mu \phi = \frac{1}{\sqrt{-g}} \partial_\mu\!\left(\sqrt{-g}\, g^{\mu\nu} \partial_\nu \phi\right)

(35)

where $g = \det(g_{\mu\nu})$ .

Part VII: Particle Motion in Curved Spacetime¶

7.1 Basic Setup¶

We study a free particle as a test body:

The particle does not back-react on the geometry
Spacetime geometry is fixed by $g_{\mu\nu}$

7.2 Worldline Parameterization¶

The particle follows a worldline $C(\lambda)$ :

x^\mu = x^\mu(\lambda), \quad \lambda \in \mathbb{R}

(36)

7.3 Proper Time¶

For timelike trajectories, define proper time $\tau$ :

d\tau^2 = - ds^2 = - g_{\mu\nu} dx^\mu dx^\nu > 0

(37)

For massive particles, we often use:

\lambda = \tau

(38)

7.4 4-Velocity¶

Definition:

U^\mu = \frac{dx^\mu}{d\tau}

(39)

Normalization:

U^\mu U_\mu = g_{\mu\nu} U^\mu U^\nu = -1

(40)

7.5 Tangent Vector¶

For general parameter $\lambda$ :

t^\mu = \frac{dx^\mu}{d\lambda}

(41)

Relation to 4-velocity:

t^\mu = \left(\frac{d\tau}{d\lambda}\right) U^\mu

(42)

Part VIII: Geodesic Equation¶

8.1 Variation Along Curves¶

For a Scalar Field $\phi$ :

\frac{d\phi}{d\lambda} = t^\mu \nabla_\mu \phi

(43)

For General Tensors $T$ :

\frac{dT}{d\lambda} = t^\mu \nabla_\mu T

(44)

8.2 4-Acceleration¶

Definition:

a^\nu = \frac{dU^\nu}{d\lambda} = t^\mu \nabla_\mu U^\nu

(45)

For $\lambda = \tau$ :

a^\nu = U^\mu \nabla_\mu U^\nu

(46)

8.3 Free Particle Condition¶

A particle is free if:

a^\nu = 0 \quad \Rightarrow \quad U^\mu \nabla_\mu U^\nu = 0

(47)

8.4 Deriving the Geodesic Equation¶

Using the covariant derivative expression:

\nabla_\mu U^\nu = \partial_\mu U^\nu + \Gamma^\nu_{\mu\alpha} U^\alpha

(48)

We obtain:

U^\mu \partial_\mu U^\nu + \Gamma^\nu_{\mu\alpha} U^\mu U^\alpha = 0

(49)

Since $U^\mu \partial_\mu = \frac{d}{d\tau}$ :

\frac{dU^\nu}{d\tau} + \Gamma^\nu_{\mu\alpha} \frac{dx^\mu}{d\tau} \frac{dx^\alpha}{d\tau} = 0

(50)

Finally, with $U^\nu = \frac{dx^\nu}{d\tau}$ :

\boxed{ \frac{d^2 x^\nu}{d\tau^2} + \Gamma^\nu_{\mu\alpha} \frac{dx^\mu}{d\tau} \frac{dx^\alpha}{d\tau} = 0 }

(51)

Part IX: Interpretation and Remarks¶

9.1 Physical Interpretation¶

Key Results:

Free particles follow geodesics of spacetime
Gravity is encoded in the connection $\Gamma^\nu_{\mu\alpha}$
In local free-falling frames ( $\Gamma^\nu_{\mu\alpha}(p) = 0$ ), motion reduces to Special Relativity

9.2 Important Remarks¶

For Massless Particles:

Proper time $\tau$ is not defined
The geodesic equation still holds using an affine parameter $\lambda$

General Principle: The geodesic equation (8.9) represents the generalization of Newton’s first law to curved spacetime.