Physlib.Mathematics.Geometry.Metric.PseudoRiemannian.Defs

Given a finite-dimensional real vector space $E$ and a quadratic form $q$ on $E$, the negative dimension $\text{negDim}(q)$ is the number of negative coefficients in the diagonalized form of $q$. Specifically, if $q$ is represented as a weighted sum of squares $\sum_{i=1}^{\dim E} w_i x_i^2$ with weights $w_i \in \{1, 0, -1\}$, then $\text{negDim}(q)$ is the count of indices $i$ such that $w_i = -1$. Geometrically, this value represents the dimension of a maximal subspace $W \subseteq E$ such that $q(v) < 0$ for all non-zero $v \in W$.

Let $E$ be a finite-dimensional vector space over $\mathbb{R}$ with dimension $n$. Let $w: \{0, \dots, n-1\} \to \mathbb{R}$ be a set of weights. For a vector $v \in \mathbb{R}^n$ defined such that its $j$-th component $v_j$ is $1$ if $j=i$ and $0$ otherwise (i.e., $v$ is the $i$-th standard basis vector $e_i$), the weighted sum of squares $\sum_{j=0}^{n-1} w_j v_j^2$ is equal to the weight $w_i$.

Let $E$ be a finite-dimensional real vector space. Let $q$ be a quadratic form on $E$ that is equivalent to a weighted sum of squares $\sum_{j=0}^{n-1} w_j x_j^2$, where $n$ is the dimension of $E$ and each weight $w_j$ is a sign in $\{-1, 0, 1\}$. If there exists an index $i$ such that the weight $w_i$ is negative, then there exists a non-zero vector $v \in E$ such that $q(v) < 0$. This provides a concrete realization of a 1-dimensional negative definite subspace contributing to the index of the form.

Let $E$ be a real vector space and let $q$ be a positive definite quadratic form on $E$. Suppose $q$ is equivalent to a weighted sum of squares $\sum_{j=1}^n w_j x_j^2$, where $n$ is the dimension of $E$ and each weight $w_j$ is a sign in $\{-1, 0, 1\}$. Then for every index $i$, the weight $w_i$ is not negative (i.e., $w_i \neq -1$).

Let $E$ be a finite-dimensional real vector space. If $q$ is a positive definite quadratic form on $E$, then its negative dimension (also known as the index of the form), denoted by $\text{negDim}(q)$, is $0$.

Let $M$ be a smooth manifold modeled on a normed space $E$, and let $T_x M$ denote the tangent space at a point $x \in M$. Given a family of continuous symmetric bilinear forms $g_x: T_x M \times T_x M \to \mathbb{R}$, this definition constructs the associated quadratic form $Q_x$ on the tangent space $T_x M$ defined by $Q_x(v) = g_x(v, v)$ for all $v \in T_x M$.

Given a pseudo-Riemannian metric $g$ on a smooth manifold $M$ and a point $x \in M$, this definition constructs the bilinear form $g_x: T_x M \times T_x M \to \mathbb{R}$ on the tangent space $T_x M$. For tangent vectors $u, v \in T_x M$, the bilinear form is defined by the value of the metric at $x$ applied to $u$ and $v$.

Given a pseudo-Riemannian metric $g$ on a smooth manifold $M$ and a point $x \in M$, this definition constructs the associated quadratic form $Q_x: T_x M \to \mathbb{R}$ on the tangent space at $x$. For any tangent vector $v \in T_x M$, the quadratic form is defined by $Q_x(v) = g_x(v, v)$, where $g_x$ is the symmetric bilinear form provided by the metric at that point.

This instance allows a pseudo-Riemannian metric $g$ on a smooth manifold $M$ to be treated as a function. Specifically, for any point $x \in M$, $g(x)$ represents a continuous bilinear form on the tangent space $T_x M$, mapping two tangent vectors $v, w \in T_x M$ to a real number.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and tangent vectors $v, w \in T_x M$, the value of the bilinear form associated with the metric at $x$, denoted by $\text{toBilinForm}(g, x)(v, w)$, is equal to the evaluation of the metric's underlying function at $x$ on $v$ and $w$, denoted by $g(x, v, w)$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any tangent vector $v \in T_x M$, the value of the quadratic form associated with the metric at $x$ evaluated at $v$ is equal to the metric's underlying bilinear form at $x$ evaluated on $(v, v)$, i.e., $\text{toQuadraticForm}(g, x)(v) = g_x(v, v)$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$, the bilinear form $g_x: T_x M \times T_x M \to \mathbb{R}$ induced by the metric on the tangent space at $x$ is symmetric.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$, the bilinear form $g_x: T_x M \times T_x M \to \mathbb{R}$ on the tangent space at $x$ is non-degenerate.

For a pseudo-Riemannian metric $g$ on a smooth manifold $M$ and a point $x \in M$, this function represents the inner product (or scalar product) of two tangent vectors $v, w \in T_x M$. The value is the real number obtained by evaluating the metric tensor at $x$ on the pair of vectors $(v, w)$, denoted as $g_x(v, w)$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any two tangent vectors $v, w \in T_x M$ in the tangent space at $x$, the inner product of $v$ and $w$ with respect to the metric $g$ at $x$ is equal to the evaluation of the underlying bilinear form of the metric tensor at $x$ on the pair $(v, w)$.

Given a pseudo-Riemannian metric $g$ on a smooth manifold $M$ and a point $x \in M$, the "flat" map $g_x^\flat$ is the linear map from the tangent space $T_x M$ to its dual space $T_x^* M$ (the space of continuous linear forms on $T_x M$). It maps a vector $v \in T_x M$ to the 1-form $g_x(v, \cdot)$, such that for any vector $w \in T_x M$, the value is given by $g_x(v, w)$. In differential geometry and physics, this is the musical isomorphism used for index lowering, which is an isomorphism due to the non-degeneracy of the pseudo-Riemannian metric.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and tangent vectors $v, w \in T_x M$, the evaluation of the 1-form $v^\flat$ (the image of $v$ under the "flat" musical isomorphism $g_x^\flat: T_x M \to T_x^* M$) on the vector $w$ is equal to the metric evaluated at $x$ on the pair $(v, w)$, which can be expressed as $(v^\flat)(w) = g_x(v, w)$.

For a pseudo-Riemannian metric $g$ on a smooth manifold $M$ and a point $x \in M$, the flat musical isomorphism $g_x^\flat$ is the continuous linear map from the tangent space $T_x M$ to the cotangent space $T_x^* M$ (the space of continuous linear forms on $T_x M$). It is defined such that for any vector $v \in T_x M$, the resulting 1-form $g_x^\flat(v)$ acts on a vector $w \in T_x M$ by $g_x(v, w)$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any tangent vectors $v, w \in T_x M$, the action of the 1-form $v^\flat$ (the image of $v$ under the continuous linear flat musical isomorphism $\flat: T_x M \to T_x^* M$) on the vector $w$ is equal to the metric evaluated at $x$ on the vectors $v$ and $w$, namely $(v^\flat)(w) = g_x(v, w)$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$ and let $x$ be a point in $M$. The "flat" musical isomorphism $g_x^\flat : T_x M \to T_x^* M$, which maps a tangent vector $v$ to the 1-form $g_x(v, \cdot)$, is injective.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$ and let $x$ be a point in $M$. The continuous linear "flat" musical isomorphism $g_x^\flat: T_x M \to T_x^* M$, which maps a tangent vector $v \in T_x M$ to the cotangent vector (1-form) $g_x(v, \cdot)$, is injective.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$ and let $x$ be a point in $M$. The continuous linear "flat" musical isomorphism $g_x^\flat: T_x M \to T_x^* M$, which maps a tangent vector $v \in T_x M$ to the cotangent vector (1-form) defined by $w \mapsto g_x(v, w)$, is surjective.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$, the "musical" isomorphism $\flat$ (flat) is the continuous linear equivalence between the tangent space $T_x M$ and the cotangent space $T_x^* M$ (defined as the space of continuous linear forms $T_x M \to \mathbb{R}$). For a vector $v \in T_x M$, the 1-form $v^\flat$ is defined by the action $v^\flat(w) = g_x(v, w)$ for all vectors $w \in T_x M$. This map is an isomorphism due to the non-degeneracy of the metric $g_x$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$, the underlying linear map of the flat musical isomorphism $\flat: T_x M \xrightarrow{\cong} T_x^* M$ (denoted by `flatEquiv`) is equal to the continuous linear map $g_x^\flat: T_x M \to T_x^* M$ (denoted by `flatL`) defined by the metric $g_x$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any tangent vectors $v, w \in T_x M$, the evaluation of the flat musical isomorphism $v^\flat$ (induced by $g$ at $x$) on the vector $w$ is equal to the metric inner product of $v$ and $w$. That is, $(v^\flat)(w) = g_x(v, w)$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$ and let $x \in M$ be a point. The "musical" isomorphism $\sharp$ (sharp) is the continuous linear equivalence from the cotangent space $T_x^* M$ (the dual space of continuous linear forms $T_x M \to \mathbb{R}$) to the tangent space $T_x M$. This map is defined as the inverse of the flat isomorphism $\flat: T_x M \xrightarrow{\cong} T_x^* M$. For any 1-form $\omega \in T_x^* M$, the vector $\omega^\sharp$ is the unique element of $T_x M$ such that $g_x(\omega^\sharp, w) = \omega(w)$ for all $w \in T_x M$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$ and let $x \in M$ be a point. The map `sharpL` is the index raising "sharp" musical isomorphism viewed as a continuous linear map from the cotangent space $T_x^* M$ (the dual space of continuous linear forms on $T_x M$) to the tangent space $T_x M$. For any 1-form $\omega \in T_x^* M$, it returns the unique vector $\omega^\sharp \in T_x M$ such that $g_x(\omega^\sharp, w) = \omega(w)$ for all $w \in T_x M$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$ and let $x \in M$ be a point. The continuous linear map $g.\text{sharpL } x: T_x^* M \to T_x M$ is equal to the continuous linear map underlying the continuous linear equivalence $g.\text{sharpEquiv } x: T_x^* M \cong T_x M$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$ and let $x \in M$ be a point. The continuous linear map induced by the sharp musical isomorphism $\text{sharpEquiv}_x : T_x^* M \cong T_x M$ is equal to the continuous linear map $\text{sharpL}_x : T_x^* M \to T_x M$. Here, $T_x M$ denotes the tangent space and $T_x^* M$ denotes the cotangent space (the space of continuous linear forms on $T_x M$).

Given a pseudo-Riemannian metric $g$ on a smooth manifold $M$ and a point $x \in M$, the map `sharp` (denoted by $\sharp$) is the linear map from the cotangent space $T_x^* M$ (the dual space of continuous linear forms on the tangent space $T_x M$) to the tangent space $T_x M$. This map is the linear map induced by the continuous linear equivalence `sharpEquiv`, and it satisfies the property that for any 1-form $\omega \in T_x^* M$, the vector $\omega^\sharp \in T_x M$ is the unique element such that $g_x(\omega^\sharp, v) = \omega(v)$ for all $v \in T_x M$.

Let $M$ be a smooth manifold equipped with a pseudo-Riemannian metric $g$. For any point $x \in M$ and any tangent vector $v \in T_x M$, let $g_x^\flat: T_x M \to T_x^* M$ be the flat musical isomorphism (which maps a vector $v$ to the 1-form $w \mapsto g_x(v, w)$) and $g_x^\sharp: T_x^* M \to T_x M$ be the sharp musical isomorphism (the inverse of $g_x^\flat$). Then, applying the sharp map to the result of the flat map on $v$ returns the original vector $v$: $$g_x^\sharp(g_x^\flat(v)) = v$$

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any 1-form $\omega$ in the cotangent space $T_x^* M$ (the dual space of the tangent space $T_x M$), applying the continuous linear sharp isomorphism $g_x^\sharp$ followed by the continuous linear flat isomorphism $g_x^\flat$ yields the original 1-form $\omega$. That is, $g_x^\flat(g_x^\sharp(\omega)) = \omega$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any 1-form (covector) $\omega$ in the cotangent space $T_x^* M$, applying the linear sharp map $g_x^\sharp: T_x^* M \to T_x M$ followed by the linear flat map $g_x^\flat: T_x M \to T_x^* M$ returns the original 1-form $\omega$. That is, $$g_x^\flat(g_x^\sharp(\omega)) = \omega$$

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any tangent vector $v$ in the tangent space $T_x M$, let $g_x^\flat: T_x M \to T_x^* M$ be the flat musical isomorphism (which maps a vector $v$ to the 1-form $g_x(v, \cdot)$) and $g_x^\sharp: T_x^* M \to T_x M$ be the sharp musical isomorphism (the inverse of the flat map). Then, applying the sharp map to the result of the flat map on $v$ returns the original vector $v$: $$g_x^\sharp(g_x^\flat(v)) = v$$

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$ and let $x \in M$ be a point. For any two 1-forms $\omega_1, \omega_2$ in the cotangent space $T_x^* M$, let $\omega_1^\sharp$ and $\omega_2^\sharp$ denote their images in the tangent space $T_x M$ under the sharp musical isomorphism $\sharp: T_x^* M \to T_x M$ induced by $g$. Then the metric evaluated at $x$ on these two vectors satisfies $g_x(\omega_1^\sharp, \omega_2^\sharp) = \omega_1(\omega_2^\sharp)$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$, any tangent vector $v \in T_x M$, and any 1-form $\omega \in T_x^* M$, the metric evaluated at $x$ on the vector $v$ and the vector $\omega^\sharp$ is equal to the evaluation of the 1-form $\omega$ on $v$. That is, $g_x(v, \omega^\sharp) = \omega(v)$, where $\omega^\sharp$ is the image of $\omega$ under the sharp musical isomorphism $\sharp: T_x^* M \to T_x M$.

Let $g$ be a pseudo-Riemannian metric on a manifold $M$ and let $x \in M$ be a point. This function defines the induced metric on the cotangent space $T_x^* M$. For any two 1-forms $\omega_1, \omega_2 \in T_x^* M$, its value is given by $g_x(\omega_1^\sharp, \omega_2^\sharp)$, where $\sharp: T_x^* M \to T_x M$ is the sharp musical isomorphism (index raising operator) induced by $g_x$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any two 1-forms $\omega_1, \omega_2 \in T_x^* M$ in the cotangent space at $x$, the value of the induced metric on the cotangent space is equal to the evaluation of the first 1-form on the "sharp" of the second 1-form: $$\text{cotangentMetricVal}(g, x, \omega_1, \omega_2) = \omega_1(\omega_2^\sharp)$$ where $\omega_2^\sharp$ is the image of $\omega_2$ under the sharp musical isomorphism $\sharp: T_x^* M \to T_x M$ induced by $g$ at $x$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any two 1-forms $\omega_1, \omega_2 \in T_x^* M$ in the cotangent space at $x$, the induced metric on the cotangent space is symmetric: $$\text{cotangentMetricVal}(g, x, \omega_1, \omega_2) = \text{cotangentMetricVal}(g, x, \omega_2, \omega_1)$$ where $\text{cotangentMetricVal}$ denotes the bilinear form on $T_x^* M$ induced by the metric $g$ at $x$.

For a pseudo-Riemannian metric $g$ on a manifold $M$, this definition constructs the induced symmetric bilinear form on the cotangent space $T_x^* M$ at a point $x \in M$. Given two covectors $\omega_1, \omega_2 \in T_x^* M$, the value of the bilinear form is given by $g_x(\omega_1^\sharp, \omega_2^\sharp)$, where $\sharp : T_x^* M \to T_x M$ is the sharp musical isomorphism (index-raising operator) induced by $g_x$.

Given a pseudo-Riemannian metric $g$ on a manifold $M$ and a point $x \in M$, this definition provides the quadratic form on the cotangent space $T_x^* M$ induced by the metric. For any 1-form $\omega \in T_x^* M$, the value of the quadratic form is given by $g_x(\omega^\sharp, \omega^\sharp)$, where $\omega^\sharp \in T_x M$ is the vector field obtained by applying the sharp musical isomorphism (index-raising operator) induced by $g_x$ to $\omega$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$ and let $x \in M$ be a point. For any two covectors $\omega_1, \omega_2$ in the cotangent space $T_x^* M$ (represented as the space of continuous linear maps $T_x M \to \mathbb{R}$), the evaluation of the induced symmetric bilinear form on the cotangent space $\text{cotangentToBilinForm}(g, x)$ at $(\omega_1, \omega_2)$ is equal to the induced metric value $\text{cotangentMetricVal}(g, x, \omega_1, \omega_2)$, which is defined as $g_x(\omega_1^\sharp, \omega_2^\sharp)$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any 1-form $\omega \in T_x^* M$, the value of the quadratic form induced by $g$ on the cotangent space at $x$, denoted as $\text{cotangentToQuadraticForm}(g, x)$, applied to $\omega$ is equal to the value of the induced metric on the cotangent space $\text{cotangentMetricVal}(g, x)$ evaluated at $(\omega, \omega)$. This is expressed by the equality: $$\text{cotangentToQuadraticForm}(g, x)(\omega) = \text{cotangentMetricVal}(g, x, \omega, \omega)$$ where $\text{cotangentMetricVal}(g, x, \omega, \omega) = g_x(\omega^\sharp, \omega^\sharp)$ and $\sharp$ is the sharp musical isomorphism induced by $g$.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$, the bilinear form induced by $g$ on the cotangent space $T_x^* M$ is symmetric.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$ and any 1-form $\omega \in T_x^* M$ in the cotangent space, if the induced metric on the cotangent space satisfies $\text{cotangentMetricVal}(g, x, \omega, v) = 0$ for all 1-forms $v \in T_x^* M$, then $\omega = 0$. This demonstrates that the induced metric on the cotangent space is non-degenerate.

Let $g$ be a pseudo-Riemannian metric on a smooth manifold $M$. For any point $x \in M$, the symmetric bilinear form induced by $g$ on the cotangent space $T_x^* M$ is non-degenerate.