18. Tensor Analysis¶

Learning Objectives¶

Understand the definition of tensors from the perspective of coordinate transformation laws and classify scalars, vectors, and matrices as special cases of tensors
Use Einstein summation convention and index notation to concisely express and manipulate tensor equations
Distinguish transformation laws for contravariant and covariant tensors and perform index raising/lowering through the metric tensor
Understand the concepts of Christoffel symbols and covariant derivatives, and correctly calculate tensor differentiation in curvilinear coordinate systems
Understand the definition and geometric meaning of the Riemann curvature tensor and calculate curvature in simple spaces
Describe physical applications of tensor analysis (stress tensor, electromagnetic field tensor, Einstein equations) and perform calculations in Python

Why are tensors necessary? Natural laws must be independent of the choice of coordinate system. Scalars (rank-0) and vectors (rank-1) alone cannot describe physical quantities such as stress, moment of inertia, and electromagnetic fields. Tensors are geometric objects that follow well-defined transformation laws under arbitrary coordinate transformations, providing a natural language to express physical laws in a coordinate-independent form.

1. Basic Concepts of Tensors¶

1.1 Motivation: Why Scalars and Vectors Are Not Enough¶

In physics, many quantities are sufficiently described by scalars (temperature, energy) or vectors (force, velocity). However, the following physical quantities require two or more directional information:

Stress tensor $\sigma_{ij}$: direction of surface ($j$) and direction of force acting on that surface ($i$)
Moment of inertia tensor $I_{ij}$: relationship between angular velocity direction and angular momentum direction
Electromagnetic field tensor $F_{\mu\nu}$: antisymmetric tensor integrating electric and magnetic fields

These are all rank-2 tensors, having $n^2$ components in $n$-dimensional space.

1.2 Coordinate Transformations and Tensor Definition¶

Under coordinate transformation $x^i \to x'^i(x^1, x^2, \ldots, x^n)$, a rank-$k$ tensor is an object whose components follow specific transformation laws:

Rank-0 (scalar): $\phi' = \phi$ (invariant)
Rank-1 (vector): $A'^i = \frac{\partial x'^i}{\partial x^j}A^j$ (for contravariant vectors)
Rank-2 tensor: $T'^{ij} = \frac{\partial x'^i}{\partial x^k}\frac{\partial x'^j}{\partial x^l}T^{kl}$

In general, tensors can be defined as multilinear maps. A rank-$(p,q)$ tensor is a multilinear function that takes $p$ covectors and $q$ contravariant vectors and produces a real number.

1.3 Tensor Rank/Order¶

Rank	Number of components ($n$-dim)	Physical examples
0	$1$	Temperature, mass, energy
1	$n$	Force, velocity, electric field
2	$n^2$	Stress, moment of inertia, metric tensor
3	$n^3$	Piezoelectric tensor
4	$n^4$	Riemann curvature tensor, elasticity tensor

1.4 Python: Coordinate Transformation Example¶

import numpy as np

# === 2차원 회전 변환에서 텐서 변환 법칙 검증 ===

# 회전 각도
theta = np.pi / 6  # 30도

# 변환 행렬: x'^i = R^i_j x^j
R = np.array([
    [np.cos(theta), np.sin(theta)],
    [-np.sin(theta), np.cos(theta)]
])
print(f"회전 행렬 R (θ = {np.degrees(theta):.0f}°):")
print(R)

# --- Rank-1 텐서 (벡터) 변환 ---
A = np.array([3.0, 4.0])  # 원래 좌표계의 벡터
A_prime = R @ A            # A'^i = R^i_j A^j
print(f"\n원래 벡터: A = {A}")
print(f"변환된 벡터: A' = {A_prime}")
print(f"|A| = {np.linalg.norm(A):.4f}, |A'| = {np.linalg.norm(A_prime):.4f}")
# 크기 보존 확인

# --- Rank-2 텐서 변환 ---
# 응력 텐서 예시
T = np.array([
    [10.0, 3.0],
    [3.0,  5.0]
])  # 대칭 텐서

# T'^{ij} = R^i_k R^j_l T^{kl} = R T R^T
T_prime = R @ T @ R.T
print(f"\n원래 텐서:\n{T}")
print(f"변환된 텐서:\n{T_prime}")

# 텐서의 불변량(trace, determinant) 확인
print(f"\ntr(T) = {np.trace(T):.4f}, tr(T') = {np.trace(T_prime):.4f}")
print(f"det(T) = {np.linalg.det(T):.4f}, det(T') = {np.linalg.det(T_prime):.4f}")
# 대각합과 행렬식은 좌표 변환에 불변

2. Index Notation and Einstein Summation Convention¶

2.1 Einstein Summation Convention¶

Einstein's summation convention is a notation that makes tensor expressions concise:

When an upper index and a lower index are repeated with the same letter in the same term, summation over that index is implied.

$$ A^i B_i \equiv \sum_{i=1}^{n} A^i B_i $$

Free index: an index that appears on both sides of an equation (represents each component of the equation)

Dummy index: a repeated index that is summed over by the summation convention (can be renamed)

$$ A^i B_i = A^j B_j \quad (\text{dummy index } i \text{ can be replaced with } j) $$

2.2 Kronecker Delta $\delta_{ij}$¶

The Kronecker delta corresponds to the components of the identity matrix:

$$ \delta_{ij} = \begin{cases} 1 & (i = j) \\ 0 & (i \ne j) \end{cases} $$

Key properties: - $\delta_{ij} A^j = A^i$ (index substitution role) - $\delta_{ii} = n$ (trace in $n$ dimensions) - $\delta_{ij}\delta_{jk} = \delta_{ik}$

2.3 Levi-Civita Symbol $\varepsilon_{ijk}$¶

The Levi-Civita symbol is a completely antisymmetric symbol:

$$ \varepsilon_{ijk} = \begin{cases} +1 & (ijk) \text{ is an even permutation of }(123) \\ -1 & (ijk) \text{ is an odd permutation of }(123) \\ 0 & \text{if any index is repeated} \end{cases} $$

Representation of cross product and determinant:

$$ (\mathbf{A} \times \mathbf{B})_i = \varepsilon_{ijk} A_j B_k $$

$$ \det(M) = \varepsilon_{ijk} M_{1i} M_{2j} M_{3k} $$

2.4 $\varepsilon$-$\delta$ Identity¶

A very useful identity in tensor calculations:

$$ \varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km} $$

From this identity, various identities such as vector triple products can be derived:

$$ \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B}) $$

2.5 Python: Tensor Operations Using numpy.einsum¶

import numpy as np

# === numpy.einsum: 아인슈타인 합산 규약의 Python 구현 ===

A = np.array([1.0, 2.0, 3.0])
B = np.array([4.0, 5.0, 6.0])

# 내적: A^i B_i
dot = np.einsum('i,i->', A, B)
print(f"내적 A·B = {dot}")  # 32.0

# 외적 (텐서곱): C^{ij} = A^i B^j
outer = np.einsum('i,j->ij', A, B)
print(f"\n외적 A⊗B:\n{outer}")

# 행렬-벡터 곱: (Mv)^i = M^{ij} v_j
M = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=float)
v = np.array([1.0, 0.0, -1.0])
Mv = np.einsum('ij,j->i', M, v)
print(f"\nMv = {Mv}")

# 행렬 곱: (AB)^{ik} = A^{ij} B^{jk}
N = np.random.rand(3, 3)
MN = np.einsum('ij,jk->ik', M, N)
print(f"\n행렬 곱 MN (einsum):\n{MN}")
print(f"행렬 곱 MN (numpy):  \n{M @ N}")

# 대각합(trace): T^i_i
trace = np.einsum('ii->', M)
print(f"\ntr(M) = {trace}")

# 이중 축약(double contraction): A_{ij} B_{ij}
A2 = np.random.rand(3, 3)
B2 = np.random.rand(3, 3)
double_contract = np.einsum('ij,ij->', A2, B2)
print(f"A:B (이중 축약) = {double_contract:.4f}")

# --- 레비-치비타 기호와 외적 ---
# 3차원 레비-치비타 기호 생성
def levi_civita_3d():
    """3차원 레비-치비타 기호 ε_{ijk}를 생성"""
    eps = np.zeros((3, 3, 3))
    eps[0, 1, 2] = eps[1, 2, 0] = eps[2, 0, 1] = 1
    eps[0, 2, 1] = eps[2, 1, 0] = eps[1, 0, 2] = -1
    return eps

eps = levi_civita_3d()

# 외적: (A × B)_i = ε_{ijk} A_j B_k
cross_einsum = np.einsum('ijk,j,k->i', eps, A, B)
cross_numpy = np.cross(A, B)
print(f"\nA × B (einsum): {cross_einsum}")
print(f"A × B (numpy):  {cross_numpy}")

# ε-δ 항등식 검증: ε_{ijk} ε_{imn} = δ_{jm}δ_{kn} - δ_{jn}δ_{km}
lhs = np.einsum('ijk,imn->jkmn', eps, eps)
delta = np.eye(3)
rhs = np.einsum('jm,kn->jkmn', delta, delta) - np.einsum('jn,km->jkmn', delta, delta)
print(f"\nε-δ 항등식 검증: {np.allclose(lhs, rhs)}")

3. Contravariant and Covariant Tensors¶

3.1 Contravariant Vector¶

Under coordinate transformation $x^i \to x'^i$, the components of a contravariant vector transform like coordinate differentials:

$$ A'^i = \frac{\partial x'^i}{\partial x^j}A^j $$

Denoted by upper indices, the displacement vector $dx^i$ is a typical example.

3.2 Covariant Vector¶

The components of a covariant vector (1-form) transform like gradients:

$$ A'_i = \frac{\partial x^j}{\partial x'^i}A_j $$

Denoted by lower indices, the partial derivative of a scalar function $\partial_i \phi = \frac{\partial \phi}{\partial x^i}$ is a typical example.

3.3 Mixed Tensors and Index Raising/Lowering¶

Mixed tensor: a tensor with both upper and lower indices. For example, a rank-$(1,1)$ tensor $T^i{}_j$:

$$ T'^i{}_j = \frac{\partial x'^i}{\partial x^k}\frac{\partial x^l}{\partial x'^j}T^k{}_l $$

Index raising/lowering is performed through the metric tensor:

$$ A^i = g^{ij}A_j \quad (\text{raising}), \qquad A_i = g_{ij}A^j \quad (\text{lowering}) $$

3.4 Python: Contravariant/Covariant Transformation Example¶

import numpy as np
import sympy as sp

# === 극좌표 ↔ 직교좌표 변환에서 반변/공변 벡터 ===

r_val, theta_val = 2.0, np.pi / 4  # (r, θ) = (2, 45°)

# 직교 좌표 → 극좌표 변환의 야코비안
# x = r cosθ, y = r sinθ
# ∂x^i/∂x'^j 계산 (x' = 극좌표, x = 직교좌표)

# ∂(x,y)/∂(r,θ) : 직교를 극좌표로 미분
J = np.array([
    [np.cos(theta_val), -r_val * np.sin(theta_val)],  # ∂x/∂r, ∂x/∂θ
    [np.sin(theta_val),  r_val * np.cos(theta_val)]   # ∂y/∂r, ∂y/∂θ
])

# 역야코비안: ∂(r,θ)/∂(x,y)
J_inv = np.linalg.inv(J)

print("야코비안 ∂(x,y)/∂(r,θ):")
print(J)
print(f"\n역야코비안 ∂(r,θ)/∂(x,y):")
print(J_inv)

# 직교좌표에서의 벡터 (반변 성분)
A_cart = np.array([1.0, 1.0])  # (Ax, Ay)

# 반변 변환: A'^i = (∂x'^i/∂x^j) A^j
# 극좌표의 반변 성분 = J_inv @ A_cart
A_polar_contra = J_inv @ A_cart
print(f"\n직교좌표 벡터 A = {A_cart}")
print(f"극좌표 반변 성분 (A^r, A^θ) = {A_polar_contra}")

# 공변 변환: A'_i = (∂x^j/∂x'^i) A_j
# 극좌표의 공변 성분 = J^T @ A_cart
A_polar_cov = J.T @ A_cart
print(f"극좌표 공변 성분 (A_r, A_θ) = {A_polar_cov}")

# 극좌표의 계량 텐서로 검증: g_{ij} = diag(1, r²)
g = np.diag([1.0, r_val**2])
A_cov_from_contra = g @ A_polar_contra
print(f"\ng_{ij} A^j = {A_cov_from_contra}")
print(f"직접 계산한 공변 성분과 일치: {np.allclose(A_polar_cov, A_cov_from_contra)}")

4. Metric Tensor¶

4.1 Line Element and Definition of Metric Tensor¶

The distance (line element) between two adjacent points in a coordinate system is defined by the metric tensor $g_{ij}$:

$$ ds^2 = g_{ij} \, dx^i \, dx^j $$

The metric tensor is a symmetric rank-2 covariant tensor that defines the inner product: $g_{ij} = g_{ji}$.

4.2 Metric Tensors in Various Coordinate Systems¶

Cartesian coordinates $(x, y, z)$:

$$ ds^2 = dx^2 + dy^2 + dz^2 \quad \Rightarrow \quad g_{ij} = \delta_{ij} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} $$

Cylindrical coordinates $(\rho, \phi, z)$:

$$ ds^2 = d\rho^2 + \rho^2 d\phi^2 + dz^2 \quad \Rightarrow \quad g_{ij} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \rho^2 & 0 \\ 0 & 0 & 1 \end{pmatrix} $$

Spherical coordinates $(r, \theta, \phi)$:

$$ ds^2 = dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta \, d\phi^2 \quad \Rightarrow \quad g_{ij} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2\sin^2\theta \end{pmatrix} $$

4.3 Metric Tensor on Surfaces¶

2D sphere ($r = R$ fixed):

$$ ds^2 = R^2 d\theta^2 + R^2 \sin^2\theta \, d\phi^2 \quad \Rightarrow \quad g_{ij} = R^2\begin{pmatrix} 1 & 0 \\ 0 & \sin^2\theta \end{pmatrix} $$

4.4 Inverse Metric Tensor and Volume Element¶

The inverse metric tensor $g^{ij}$ satisfies $g^{ik}g_{kj} = \delta^i{}_j$.

Volume element: The volume element of a coordinate system is determined by the determinant of the metric tensor:

$$ dV = \sqrt{|g|} \, dx^1 \, dx^2 \cdots dx^n, \quad g = \det(g_{ij}) $$

4.5 Python: Metric Tensor Calculation¶

import sympy as sp

# === 다양한 좌표계에서 계량 텐서 계산 ===

def compute_metric(coords, transform):
    """
    좌표 변환으로부터 계량 텐서를 계산한다.

    Parameters:
        coords: 곡선좌표 변수 리스트
        transform: 직교좌표 표현 리스트 [x(...), y(...), z(...)]
    Returns:
        계량 텐서 행렬 (SymPy Matrix)
    """
    n = len(coords)
    r = sp.Matrix(transform)
    g = sp.zeros(n, n)
    for i in range(n):
        for j in range(i, n):
            g[i, j] = sp.trigsimp(r.diff(coords[i]).dot(r.diff(coords[j])))
            g[j, i] = g[i, j]
    return g

# 구면 좌표
r, theta, phi = sp.symbols('r theta phi', positive=True)
g_sph = compute_metric(
    [r, theta, phi],
    [r * sp.sin(theta) * sp.cos(phi),
     r * sp.sin(theta) * sp.sin(phi),
     r * sp.cos(theta)]
)
print("구면 좌표 계량 텐서:")
sp.pprint(g_sph)
print(f"det(g) = {sp.trigsimp(g_sph.det())}")
print(f"sqrt(|g|) = {sp.sqrt(sp.trigsimp(g_sph.det()))}")
# r^4 sin^2(theta) → sqrt = r^2 sin(theta)

# 2차원 구면 (r = R 고정)
R = sp.Symbol('R', positive=True)
g_sphere = compute_metric(
    [theta, phi],
    [R * sp.sin(theta) * sp.cos(phi),
     R * sp.sin(theta) * sp.sin(phi),
     R * sp.cos(theta)]
)
print("\n2차원 구면 계량 텐서:")
sp.pprint(g_sphere)

# 역계량 텐서
g_sph_inv = g_sph.inv()
print("\n구면 좌표 역계량 텐서 g^{ij}:")
sp.pprint(sp.simplify(g_sph_inv))

# 검증: g^{ik} g_{kj} = δ^i_j
identity_check = sp.simplify(g_sph_inv * g_sph)
print(f"\ng^{{ik}} g_{{kj}} = I 검증: {identity_check == sp.eye(3)}")

5. Tensor Algebra¶

5.1 Tensor Addition and Scalar Multiplication¶

Only tensors of the same type can be added:

$$ (A + B)^{ij} = A^{ij} + B^{ij}, \quad (\alpha A)^{ij} = \alpha A^{ij} $$

5.2 Tensor Product (Outer Product)¶

The tensor product of a rank-$(p,q)$ tensor and a rank-$(r,s)$ tensor produces a rank-$(p+r, q+s)$ tensor:

$$ (A \otimes B)^{ij}{}_{kl} = A^i{}_k \, B^j{}_l $$

5.3 Contraction¶

Setting one upper index and one lower index to the same letter and summing reduces the rank by 2:

$$ T^i{}_{ij} = \sum_i T^i{}_{ij} \quad (\text{rank-}(2,1) \to \text{rank-}(1,0)) $$

Typical example: trace $T^i{}_i = \text{tr}(T)$

5.4 Symmetrization and Antisymmetrization¶

Symmetric tensor: $T_{ij} = T_{ji}$

$$ T_{(ij)} = \frac{1}{2}(T_{ij} + T_{ji}) \quad (\text{symmetrization}) $$

Antisymmetric tensor: $T_{ij} = -T_{ji}$

$$ T_{[ij]} = \frac{1}{2}(T_{ij} - T_{ji}) \quad (\text{antisymmetrization}) $$

Any rank-2 tensor can be decomposed into symmetric and antisymmetric parts:

$$ T_{ij} = T_{(ij)} + T_{[ij]} $$

5.5 Python: Tensor Algebra Operations¶

import numpy as np

# === 텐서 대수 연산 ===

# rank-2 텐서 (3×3)
T = np.array([
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
], dtype=float)

# 대칭 부분과 반대칭 부분 분해
T_sym = 0.5 * (T + T.T)       # T_{(ij)}
T_antisym = 0.5 * (T - T.T)   # T_{[ij]}
print("원래 텐서 T:")
print(T)
print("\n대칭 부분 T_{(ij)}:")
print(T_sym)
print("\n반대칭 부분 T_{[ij]}:")
print(T_antisym)
print(f"\n복원 검증: T = T_sym + T_antisym? {np.allclose(T, T_sym + T_antisym)}")

# 텐서곱 (outer product)
A = np.array([1, 2, 3], dtype=float)
B = np.array([4, 5, 6], dtype=float)
AB_outer = np.einsum('i,j->ij', A, B)  # A^i B^j
print(f"\n텐서곱 A⊗B:\n{AB_outer}")

# 축약 (contraction)
# rank-2 텐서의 trace: T^i_i
trace_T = np.einsum('ii->', T)
print(f"\n축약 (trace): T^i_i = {trace_T}")

# rank-4 텐서에서 축약
R = np.random.rand(3, 3, 3, 3)
# R^i_{jkl} → 첫째와 셋째 인덱스 축약 → R^i_{jil} = S_{jl}
S = np.einsum('ijil->jl', R)
print(f"\nrank-4 텐서 축약 결과 (rank-2): shape = {S.shape}")

6. Covariant Derivative and Christoffel Symbols¶

6.1 Problems with Ordinary Differentiation¶

In curvilinear coordinate systems, the ordinary partial derivative of a tensor $\partial_i A^j$ is not a tensor. This is because the basis vectors themselves vary from point to point. A proper differential operator is needed, which is the covariant derivative.

6.2 Definition of Christoffel Symbols¶

The Christoffel symbol of the second kind $\Gamma^k{}_{ij}$ is calculated from the metric tensor:

$$ \Gamma^k{}_{ij} = \frac{1}{2}g^{kl}\left(\frac{\partial g_{jl}}{\partial x^i} + \frac{\partial g_{il}}{\partial x^j} - \frac{\partial g_{ij}}{\partial x^l}\right) $$

Christoffel symbols are not tensors (they follow inhomogeneous transformation laws). They serve to correct for "non-inertial" effects of the coordinate system.

Symmetry: $\Gamma^k{}_{ij} = \Gamma^k{}_{ji}$ (when the metric tensor is torsion-free)

6.3 Covariant Derivative¶

Covariant derivative of a contravariant vector:

$$ \nabla_i A^j = \partial_i A^j + \Gamma^j{}_{ik} A^k $$

Covariant derivative of a covariant vector:

$$ \nabla_i A_j = \partial_i A_j - \Gamma^k{}_{ij} A_k $$

Covariant derivative of a general tensor (add $+\Gamma$ for each upper index, $-\Gamma$ for each lower index):

$$ \nabla_i T^j{}_k = \partial_i T^j{}_k + \Gamma^j{}_{il} T^l{}_k - \Gamma^l{}_{ik} T^j{}_l $$

Covariant derivative of the metric tensor: $\nabla_i g_{jk} = 0$ (metric compatibility condition)

6.4 Parallel Transport and Geodesic Equation¶

The condition for parallel transport of a vector $A^i$ along a curve $x^i(\tau)$:

$$ \frac{DA^i}{D\tau} = \frac{dA^i}{d\tau} + \Gamma^i{}_{jk}\frac{dx^j}{d\tau}A^k = 0 $$

A geodesic is a curve that parallel transports its own tangent vector:

$$ \frac{d^2x^k}{d\tau^2} + \Gamma^k{}_{ij}\frac{dx^i}{d\tau}\frac{dx^j}{d\tau} = 0 $$

In flat space, geodesics are straight lines; on a sphere, they are great circles.

6.5 Python: Christoffel Symbols and Geodesic Calculation¶

import sympy as sp

# === 크리스토펠 기호 계산 함수 ===

def christoffel_symbols(g, coords):
    """
    계량 텐서로부터 크리스토펠 기호 Γ^k_{ij}를 계산한다.

    Parameters:
        g: 계량 텐서 (SymPy Matrix)
        coords: 좌표 변수 리스트
    Returns:
        Γ[k][i][j] 형태의 3차원 리스트
    """
    n = len(coords)
    g_inv = g.inv()

    Gamma = [[[sp.Integer(0) for _ in range(n)]
              for _ in range(n)]
             for _ in range(n)]

    for k in range(n):
        for i in range(n):
            for j in range(n):
                s = sp.Integer(0)
                for l in range(n):
                    s += sp.Rational(1, 2) * g_inv[k, l] * (
                        sp.diff(g[j, l], coords[i]) +
                        sp.diff(g[i, l], coords[j]) -
                        sp.diff(g[i, j], coords[l])
                    )
                Gamma[k][i][j] = sp.simplify(s)
    return Gamma

# --- 구면 좌표에서 크리스토펠 기호 ---
r, theta, phi = sp.symbols('r theta phi', positive=True)
g_sph = sp.diag(1, r**2, r**2 * sp.sin(theta)**2)
coords_sph = [r, theta, phi]

Gamma_sph = christoffel_symbols(g_sph, coords_sph)

print("구면 좌표의 0이 아닌 크리스토펠 기호:")
names = ['r', 'θ', 'φ']
for k in range(3):
    for i in range(3):
        for j in range(i, 3):
            if Gamma_sph[k][i][j] != 0:
                print(f"  Γ^{names[k]}_{{{names[i]}{names[j]}}} = {Gamma_sph[k][i][j]}")

# --- 2차원 구면 위의 측지선 (수치 계산) ---
import numpy as np
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt

def geodesic_sphere(tau, y, R_val=1.0):
    """
    단위 구면 위의 측지선 방정식.
    y = [θ, φ, dθ/dτ, dφ/dτ]
    """
    th, ph, dth, dph = y

    # 2차원 구면의 크리스토펠 기호
    # Γ^θ_{φφ} = -sinθ cosθ
    # Γ^φ_{θφ} = Γ^φ_{φθ} = cosθ/sinθ

    d2th = np.sin(th) * np.cos(th) * dph**2
    d2ph = -2.0 * np.cos(th) / np.sin(th) * dth * dph if np.sin(th) > 1e-10 else 0.0

    return [dth, dph, d2th, d2ph]

# 초기 조건: 적도에서 북동 방향으로 출발
th0, ph0 = np.pi / 2, 0.0
dth0, dph0 = -0.3, 1.0  # 북쪽으로 약간 + 동쪽으로

y0 = [th0, ph0, dth0, dph0]
sol = solve_ivp(geodesic_sphere, [0, 8], y0, max_step=0.01, dense_output=True)

# 구면 위에 측지선 표시
fig = plt.figure(figsize=(8, 8))
ax = fig.add_subplot(111, projection='3d')

# 구면 그리기
u = np.linspace(0, np.pi, 40)
v = np.linspace(0, 2*np.pi, 40)
U, V = np.meshgrid(u, v)
X = np.sin(U) * np.cos(V)
Y = np.sin(U) * np.sin(V)
Z = np.cos(U)
ax.plot_surface(X, Y, Z, alpha=0.15, color='lightblue')

# 측지선 (대원)
th_geo = sol.y[0]
ph_geo = sol.y[1]
xg = np.sin(th_geo) * np.cos(ph_geo)
yg = np.sin(th_geo) * np.sin(ph_geo)
zg = np.cos(th_geo)
ax.plot(xg, yg, zg, 'r-', linewidth=2.5, label='측지선 (대원)')
ax.plot([xg[0]], [yg[0]], [zg[0]], 'go', markersize=8, label='출발점')

ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
ax.set_title('구면 위의 측지선')
ax.legend()
plt.tight_layout()
plt.savefig('geodesic_sphere.png', dpi=150, bbox_inches='tight')
plt.show()

7. Curvature Tensor¶

7.1 Riemann Curvature Tensor¶

The Riemann curvature tensor $R^l{}_{kij}$ is defined by the non-commutativity of covariant derivatives:

$$ (\nabla_i \nabla_j - \nabla_j \nabla_i)A^l = R^l{}_{kij}A^k $$

Explicit expression using Christoffel symbols:

$$ R^l{}_{kij} = \partial_i \Gamma^l{}_{jk} - \partial_j \Gamma^l{}_{ik} + \Gamma^l{}_{im}\Gamma^m{}_{jk} - \Gamma^l{}_{jm}\Gamma^m{}_{ik} $$

7.2 Geometric Meaning¶

The Riemann curvature tensor measures the extent to which a vector changes when parallel transported along a closed path.

Flat space: $R^l{}_{kij} = 0$ (parallel transport is path-independent)
Curved space: $R^l{}_{kij} \ne 0$ (path-dependent)

7.3 Ricci Tensor, Scalar Curvature, and Einstein Tensor¶

Ricci tensor: contraction of the Riemann tensor

$$ R_{ij} = R^k{}_{ikj} $$

Scalar curvature (Ricci scalar): contraction of the Ricci tensor

$$ R = g^{ij}R_{ij} $$

Gaussian curvature on 2D surfaces:

$$ K = \frac{R}{2} \quad (\text{in 2D}) $$

Sphere ($r = a$): $K = 1/a^2 > 0$
Plane: $K = 0$
Hyperboloid: $K < 0$

Einstein tensor:

$$ G_{ij} = R_{ij} - \frac{1}{2}g_{ij}R $$

$G_{ij}$ satisfies $\nabla_i G^{ij} = 0$ (by Bianchi identities), which is directly related to energy-momentum conservation.

7.4 Python: Curvature Calculation¶

import sympy as sp

# === 리만 곡률 텐서 계산 함수 ===

def riemann_tensor(Gamma, coords):
    """
    크리스토펠 기호로부터 리만 곡률 텐서 R^l_{kij}를 계산한다.
    """
    n = len(coords)
    R = [[[[sp.Integer(0) for _ in range(n)]
           for _ in range(n)]
          for _ in range(n)]
         for _ in range(n)]

    for l in range(n):
        for k in range(n):
            for i in range(n):
                for j in range(n):
                    # ∂_i Γ^l_{jk} - ∂_j Γ^l_{ik}
                    term = sp.diff(Gamma[l][j][k], coords[i]) - \
                           sp.diff(Gamma[l][i][k], coords[j])
                    # + Γ^l_{im} Γ^m_{jk} - Γ^l_{jm} Γ^m_{ik}
                    for m in range(n):
                        term += Gamma[l][i][m] * Gamma[m][j][k] - \
                                Gamma[l][j][m] * Gamma[m][i][k]
                    R[l][k][i][j] = sp.simplify(term)
    return R

def ricci_tensor(R_riem, n):
    """리치 텐서 R_{ij} = R^k_{ikj}"""
    Ric = sp.zeros(n, n)
    for i in range(n):
        for j in range(n):
            s = sp.Integer(0)
            for k in range(n):
                s += R_riem[k][i][k][j]
            Ric[i, j] = sp.simplify(s)
    return Ric

def scalar_curvature(Ric, g_inv, n):
    """스칼라 곡률 R = g^{ij} R_{ij}"""
    R_scalar = sp.Integer(0)
    for i in range(n):
        for j in range(n):
            R_scalar += g_inv[i, j] * Ric[i, j]
    return sp.simplify(R_scalar)

# --- 2차원 구면 (r = a)의 가우스 곡률 계산 ---
theta, phi = sp.symbols('theta phi', positive=True)
a = sp.Symbol('a', positive=True)

g_sphere = sp.diag(a**2, a**2 * sp.sin(theta)**2)
coords_sphere = [theta, phi]

# 크리스토펠 기호
Gamma_sp = christoffel_symbols(g_sphere, coords_sphere)
print("2차원 구면의 0이 아닌 크리스토펠 기호:")
snames = ['θ', 'φ']
for k in range(2):
    for i in range(2):
        for j in range(i, 2):
            if Gamma_sp[k][i][j] != 0:
                print(f"  Γ^{snames[k]}_{{{snames[i]}{snames[j]}}} = {Gamma_sp[k][i][j]}")

# 리만 텐서
R_riem_sp = riemann_tensor(Gamma_sp, coords_sphere)

# 리치 텐서
Ric_sp = ricci_tensor(R_riem_sp, 2)
print(f"\n리치 텐서 R_{{ij}}:")
sp.pprint(Ric_sp)

# 스칼라 곡률
g_sp_inv = g_sphere.inv()
R_sc = scalar_curvature(Ric_sp, g_sp_inv, 2)
print(f"\n스칼라 곡률 R = {R_sc}")

# 가우스 곡률 (2차원에서 K = R/2)
K = sp.simplify(R_sc / 2)
print(f"가우스 곡률 K = R/2 = {K}")
# 출력: K = 1/a^2 (양의 일정한 곡률 → 구면)

8. Physical Applications¶

8.1 Continuum Mechanics: Stress Tensor¶

The Cauchy stress tensor $\sigma_{ij}$ describes the stress acting on any surface within a continuum. When the normal direction of the surface is $\hat{n}$, the force per unit area (traction) acting on that surface is:

$$ t_i = \sigma_{ij} n_j $$

$\sigma_{ij}$ is a symmetric tensor ($\sigma_{ij} = \sigma_{ji}$, by conservation of angular momentum), with diagonal components representing normal stress and off-diagonal components representing shear stress.

import numpy as np
import matplotlib.pyplot as plt

# === 2D 응력 텐서의 모어 원 (Mohr's Circle) ===
# 응력 텐서 σ = [[σ_xx, τ_xy], [τ_xy, σ_yy]]
sigma_xx, sigma_yy, tau_xy = 50.0, 20.0, 15.0
sigma = np.array([[sigma_xx, tau_xy],
                   [tau_xy, sigma_yy]])

# 주응력 (고유값)
eigenvalues, eigenvectors = np.linalg.eigh(sigma)
sigma_1 = eigenvalues[1]  # 최대 주응력
sigma_2 = eigenvalues[0]  # 최소 주응력
print(f"주응력: σ₁ = {sigma_1:.2f} MPa, σ₂ = {sigma_2:.2f} MPa")
print(f"주응력 방향:\n{eigenvectors}")

# 모어 원 그리기
center = (sigma_1 + sigma_2) / 2
radius = (sigma_1 - sigma_2) / 2

fig, ax = plt.subplots(figsize=(8, 6))
circle = plt.Circle((center, 0), radius, fill=False, color='blue', linewidth=2)
ax.add_patch(circle)

# 원래 응력 상태 표시
ax.plot(sigma_xx, tau_xy, 'ro', markersize=8, label=f'(σ_xx, τ_xy) = ({sigma_xx}, {tau_xy})')
ax.plot(sigma_yy, -tau_xy, 'go', markersize=8, label=f'(σ_yy, -τ_xy) = ({sigma_yy}, {-tau_xy})')
ax.plot([sigma_1, sigma_2], [0, 0], 'k^', markersize=10, label=f'주응력 σ₁={sigma_1:.1f}, σ₂={sigma_2:.1f}')

ax.axhline(y=0, color='gray', linestyle='-', alpha=0.3)
ax.axvline(x=0, color='gray', linestyle='-', alpha=0.3)
ax.set_xlabel('법선 응력 σ (MPa)')
ax.set_ylabel('전단 응력 τ (MPa)')
ax.set_title('모어 원 (Mohr\'s Circle)')
ax.set_aspect('equal')
ax.legend(fontsize=9)
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('mohr_circle.png', dpi=150, bbox_inches='tight')
plt.show()

8.2 Electromagnetism: Electromagnetic Field Tensor $F_{\mu\nu}$¶

In special relativistic electromagnetism, the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ are unified into a single antisymmetric rank-2 tensor $F_{\mu\nu}$:

$$ F_{\mu\nu} = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{pmatrix} $$

Tensor form of Maxwell's equations:

$$ \partial_\mu F^{\mu\nu} = \mu_0 J^\nu \quad (\text{inhomogeneous equations: Gauss's law + Ampère-Maxwell law}) $$

$$ \partial_{[\lambda} F_{\mu\nu]} = 0 \quad (\text{homogeneous equations: Faraday's law + magnetic Gauss's law}) $$

This representation makes covariance under Lorentz transformations manifest.

import numpy as np

# === 전자기장 텐서 구성 및 로렌츠 변환 ===

c = 1.0  # 자연 단위 (c = 1)

def em_field_tensor(E, B):
    """전기장 E와 자기장 B로부터 전자기장 텐서 F_μν를 구성한다."""
    Ex, Ey, Ez = E
    Bx, By, Bz = B
    F = np.array([
        [0,      -Ex/c,  -Ey/c,  -Ez/c],
        [Ex/c,    0,     -Bz,     By  ],
        [Ey/c,    Bz,     0,     -Bx  ],
        [Ez/c,   -By,     Bx,     0   ]
    ])
    return F

# x 방향 전기장만 있는 경우
E = np.array([1.0, 0.0, 0.0])
B = np.array([0.0, 0.0, 0.0])
F = em_field_tensor(E, B)
print("전자기장 텐서 F_μν (순수 전기장):")
print(F)

# 로렌츠 변환 (x 방향, 속도 v = 0.6c)
v = 0.6
gamma = 1.0 / np.sqrt(1 - v**2)
beta = v

# 로렌츠 변환 행렬 Λ^μ'_ν
Lambda = np.array([
    [gamma,      -gamma*beta, 0, 0],
    [-gamma*beta, gamma,      0, 0],
    [0,           0,          1, 0],
    [0,           0,          0, 1]
])

# 텐서 변환: F'^{μν} = Λ^μ_α Λ^ν_β F^{αβ}
# 먼저 F^{μν} = η^{μα} η^{νβ} F_{αβ} (민코프스키 계량으로 인덱스 올림)
eta = np.diag([-1, 1, 1, 1])  # 민코프스키 계량 (-,+,+,+)
F_up = eta @ F @ eta           # F^{μν}

F_prime_up = Lambda @ F_up @ Lambda.T
F_prime = eta @ F_prime_up @ eta  # F'_{μν}로 내림

print(f"\n로렌츠 변환 후 (v = {v}c):")
print(f"F'_μν:")
print(np.round(F_prime, 4))

# 변환된 전기장과 자기장 추출
E_prime = np.array([F_prime[0, 1], F_prime[0, 2], F_prime[0, 3]]) * (-c)
B_prime = np.array([F_prime[2, 3], F_prime[3, 1], F_prime[1, 2]])
print(f"\n변환된 전기장: E' = {np.round(E_prime, 4)}")
print(f"변환된 자기장: B' = {np.round(B_prime, 4)}")
print("순수 전기장이 로렌츠 변환에 의해 자기장 성분을 획득함!")

# 로렌츠 불변량 검증
inv1 = -0.5 * np.einsum('ij,ij->', F, F)   # F_{μν}F^{μν}/2
inv1_prime = -0.5 * np.einsum('ij,ij->', F_prime, F_prime)
print(f"\n로렌츠 불변량 F_μν F^μν: 원래 = {inv1:.4f}, 변환 후 = {inv1_prime:.4f}")

8.3 General Relativity: Einstein Field Equations¶

The core of general relativity, the Einstein field equations, connects the geometry of spacetime (curvature) with matter-energy distribution:

$$ G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu} $$

Where: - $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$: Einstein tensor (geometry) - $\Lambda$: cosmological constant - $T_{\mu\nu}$: energy-momentum tensor (matter)

Schwarzschild metric: spherically symmetric vacuum solution

$$ ds^2 = -\left(1 - \frac{r_s}{r}\right)c^2 dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1}dr^2 + r^2 d\theta^2 + r^2\sin^2\theta \, d\phi^2 $$

where $r_s = 2GM/c^2$ is the Schwarzschild radius.

import sympy as sp

# === 슈바르츠실트 계량의 크리스토펠 기호 계산 ===
t, r, theta, phi = sp.symbols('t r theta phi')
r_s, c_sym = sp.symbols('r_s c', positive=True)

# 슈바르츠실트 계량 텐서 (대각)
f = 1 - r_s / r  # f(r) = 1 - r_s/r

g_schw = sp.diag(
    -f * c_sym**2,   # g_{tt}
    1 / f,           # g_{rr}
    r**2,            # g_{θθ}
    r**2 * sp.sin(theta)**2  # g_{φφ}
)
coords_schw = [t, r, theta, phi]

print("슈바르츠실트 계량 텐서:")
sp.pprint(g_schw)

# 크리스토펠 기호 계산 (시간 소요 가능)
Gamma_schw = christoffel_symbols(g_schw, coords_schw)

print("\n슈바르츠실트 계량의 0이 아닌 크리스토펠 기호:")
coord_names = ['t', 'r', 'θ', 'φ']
for k in range(4):
    for i in range(4):
        for j in range(i, 4):
            val = Gamma_schw[k][i][j]
            if val != 0:
                print(f"  Γ^{coord_names[k]}_{{{coord_names[i]}{coord_names[j]}}} = {val}")

Practice Problems¶

Problem 1: Tensor Transformation¶

Given the 2D coordinate transformation $x' = x\cosh\alpha + y\sinh\alpha$, $y' = x\sinh\alpha + y\cosh\alpha$ (Lorentz boost):

(a) Find the transformation matrix $\frac{\partial x'^i}{\partial x^j}$.

(b) Transform the vector $A^i = (3, 4)$ to the new coordinate system ($\alpha = 0.5$).

Problem 2: Einstein Summation Convention¶

Write the following tensor expressions using Einstein summation convention (without summation symbols) and expand them in 3D:

(a) Dot product of vectors $\sum_i A^i B_i$

(b) Matrix multiplication $\sum_k M^i{}_k N^k{}_j$

Problem 3: Metric Tensor and Index Raising/Lowering¶

In 2D polar coordinates $(r, \theta)$ with $ds^2 = dr^2 + r^2 d\theta^2$:

(a) Find the metric tensor $g_{ij}$ and inverse metric tensor $g^{ij}$.

(b) Find the covariant components $A_i = g_{ij}A^j$ of the contravariant vector $A^i = (A^r, A^\theta) = (2, 1/r)$.

Problem 4: Christoffel Symbols¶

For the 2D polar coordinate metric tensor $g = \text{diag}(1, r^2)$:

(a) Calculate all Christoffel symbols $\Gamma^k{}_{ij}$.

(b) Calculate the covariant divergence $\nabla_i A^i$ of the vector field $A^r = \cos\theta$, $A^\theta = -\sin\theta / r$.

Problem 5: Geodesics¶

The metric of a 2D pseudosphere ($K = -1$) is given by $ds^2 = du^2 + e^{-2u}dv^2$.

(a) Find the Christoffel symbols.

(b) Calculate the Gaussian curvature $K$ and verify that $K = -1$.

Problem 6: Curvature Tensor¶

The metric of a torus surface is given in parameters $(\theta, \phi)$ as:

$$ ds^2 = a^2 d\theta^2 + (R + a\cos\theta)^2 d\phi^2 $$

where $R$ is the major radius and $a$ is the minor radius.

(a) Find the Gaussian curvature $K(\theta)$.

(b) Classify regions where $K > 0$, $K = 0$, and $K < 0$ by $\theta$ values.

Problem 7: Electromagnetic Field Tensor¶

Given electric field $\mathbf{E} = E_0 \hat{x}$ and magnetic field $\mathbf{B} = B_0 \hat{z}$:

(a) Construct the electromagnetic field tensor $F_{\mu\nu}$.

(b) Calculate the Lorentz invariants $F_{\mu\nu}F^{\mu\nu}$ and $\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$.

Problem 8: Principal Axes of Stress Tensor¶

A 3D stress tensor is given as:

$$ \sigma_{ij} = \begin{pmatrix} 100 & 30 & 0 \\ 30 & 50 & 20 \\ 0 & 20 & 80 \end{pmatrix} \text{ (MPa)} $$

(a) Find the principal stresses and principal directions (eigenvalues/eigenvectors).

(b) Find the maximum shear stress.

(c) Calculate the von Mises stress: $\sigma_v = \sqrt{\frac{1}{2}[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2]}$.

Advanced Topics¶

Differential Forms¶

The modern language of tensor analysis, differential forms, systematically handles antisymmetric covariant tensors. Using exterior algebra and the exterior derivative $d$:

0-form = scalar function
1-form = $\omega = A_i dx^i$
2-form = $F = \frac{1}{2}F_{ij}dx^i \wedge dx^j$

Maxwell's equations become extremely concise in differential forms: $dF = 0$, $d{*F} = J$

Lie Derivative¶

The Lie derivative $\mathcal{L}_X T$ defines the change of a tensor $T$ along a vector field $X$ in a coordinate-independent way. This is a key tool for handling symmetries and conservation laws (Killing vector fields).

Fiber Bundles and Gauge Theory¶

In Yang-Mills theory, the gauge connection is a generalization of Christoffel symbols, and curvature corresponds to the field strength tensor. This forms the mathematical foundation of the Standard Model.

Computational Tools¶

Tool	Language	Description
`sympy.diffgeom`	Python	Differential geometry calculations
`einsteinpy`	Python	General relativity tensor calculations
`xAct` / `xTensor`	Mathematica	Professional tensor algebra system
`SageManifolds`	SageMath	Differential manifold calculations
`cadabra2`	Python/C++	Field theory tensor calculations

Recommended References¶

Boas, M. L. (2005). Mathematical Methods in the Physical Sciences, 3rd ed., Chapter 10. Wiley.
Carroll, S. M. (2019). Spacetime and Geometry, 2nd ed. Cambridge University Press.
The best textbook for tensor analysis in general relativity
Schutz, B. (2009). A First Course in General Relativity, 2nd ed. Cambridge University Press.
Arfken, G. B., Weber, H. J., & Harris, F. E. (2012). Mathematical Methods for Physicists, 7th ed., Chapter 3. Academic Press.
Nakahara, M. (2003). Geometry, Topology and Physics, 2nd ed. CRC Press.
Mathematical foundations of differential forms, fiber bundles, and gauge theory

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