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# Exponentially Weighted Metric Factors and Tensor Index Transformations

Wen-Xiang Chen

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## To cite this version:

Wen-Xiang Chen. Exponentially Weighted Metric Factors and Tensor Index Transformations. 2025. [hal-05185436]

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Preprint submitted on 25 Jul 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Exponentially Weighted Metric Factors and Tensor
Index Transformations

Wen-Xiang Chen

School of Physics and Materials Science
Guangzhou University

wxchen4277@qq.com

July 24, 2025

## Abstract

In tensor analysis the raising and lowering of indices is performed using the metric tensor. This paper provides a rigorous derivation showing how a multiplicative exponential factor arising from a scalar function can be used to encapsulate the relationship between contravariant and covariant components. By introducing a scalar function $f(x, y, z) = z - \frac{x}{y}$ for algebraic variables $x, y, z$, we show that the associated metric can be written with an exponential scale factor. The properties of the exponential function and the definition of contravariant and covariant components are used to prove an equivalence between the traditional index transformation and the product of exponential factors $\exp(-x/y)$ and $\exp(z)$. This demonstration clarifies how index manipulations can be interpreted as conformal scalings of the underlying metric and offers insight into the role of scalar fields in conformal geometry.

## 1 Introduction

Tensors are fundamental objects in mathematics and physics. They are defined independent of coordinate systems but their components depend on the basis chosen to represent them. The distinction between contravariant components $T^i$ and covariant components $T_i$ of a tensor is most easily expressed using the metric tensor $g_{ij}$. In a coordinate basis the operation of lowering an index multiplies the contravariant component by $g_{ij}$ and sums over the index, while raising an index multiplies by the inverse metric $g^{ij}$ and contracts with the covariant component. These operations are inverse to each other and satisfy $g^{ij}g_{jk} = \delta^i_k$, where $\delta^i_k$ is the Kronecker delta. The exponential function, on the other hand, converts sums into products according to the functional equation $\exp(x + y) = \exp(x) \exp(y)$. When a metric is rescaled by an exponential of a scalar function, the raising and lowering of indices introduces multiplicative exponential factors. The goal of this work is to provide a detailed derivation showing that for a suitable choice of scalar function the exponential factor $\exp(z) \exp(-x/y)$ exactly encodes the transformation between the contravariant and covariant components of a vector in an appropriately scaled metric.
# 2 Background

## 2.1 Contravariant and covariant components

Let $V$ be a vector space with basis $\{e_i\}$ and dual basis $\{e^i\}$. A vector $v$ can be expressed as $v = v^i e_i$, where $v^i$ are contravariant components, or using the dual basis as $v = v_i e^i$. The metric tensor $g_{ij}$ encodes the inner product $g_{ij} = e_i \cdot e_j$. The inverse metric $g^{ij}$ satisfies $g_{ij}g^{jk} = \delta_i^k$. Given a contravariant vector $v^j$, its covariant form is obtained by lowering the index: [2]

$$v_i = g_{ij}v^j, \qquad (1)$$

and conversely raising the index returns the original vector component:

$$v^i = g^{ij}v_j. \qquad (2)$$

These operations are inverse because contracting the metric and its inverse gives the identity $\delta_i^j$. The fundamental role of the metric in index manipulations is emphasized in tensor analysis texts. For example, reference [1] notes that “a contravariant vector can be lowered (converted to a covariant vector) by multiplying it by $g_{ij}$, and the covariant vector can be converted back into a contravariant vector by raising its index”. A similar description is found in general relativity notes, where the metric and its inverse are used to raise and lower indices of tensors of any order.

## 2.2 Properties of the exponential function

The real exponential function $\exp(x)$ is characterized by the property $\exp(x + y) = \exp(x) \exp(y)$. This functional equation implies that the exponential maps sums into products, while the inverse function, the natural logarithm, maps products into sums. The same property holds for negative arguments: $\exp(-x) = 1/\exp(x)$. These identities will be crucial when constructing an exponential scale factor that reproduces the effect of index lowering via a conformal rescaling of the metric. In particular, for any real numbers $a$ and $b$, the identity [3]

$$\exp(a - b) = \exp(a) \exp(-b) \qquad (3)$$

follows directly from the functional equation and the reciprocal property of the exponential function.

# 3 Exponentially rescaled metrics

To connect index transformations with exponential factors we consider a conformal rescaling of a base metric. Let $\gamma_{ij}$ denote a fixed metric (for example the Euclidean metric $\delta_{ij}$) and let $f(x, y, z)$ be a scalar function of algebraic variables $x, y$ and $z$. We define a new metric

$$g_{ij}(x, y, z) = e^{2f(x, y, z)} \gamma_{ij}. \qquad (4)$$

Since the factor $e^{2f}$ is positive, $g_{ij}$ remains a valid metric. The inverse metric is obtained by dividing by the same factor:

$$g^{ij}(x, y, z) = e^{-2f(x, y, z)} \gamma^{ij}, \qquad (5)$$
where $\gamma^{ij}$ is the inverse of the base metric. The product $g^{ij}g_{jk} = \delta^i_k$ is preserved because the exponential factors cancel. If $v^i$ denotes the contravariant component of a vector with respect to $\gamma_{ij}$, then its covariant component with respect to the rescaled metric is

$$v_i = g_{ij}v^j = e^{2f(x,y,z)}\gamma_{ij}v^j. \qquad (6)$$

Conversely, raising the index using $g^{ij}$ divides by the exponential factor:

$$v^i = g^{ij}v_j = e^{-2f(x,y,z)}\gamma^{ij}v_j. \qquad (7)$$

Thus, for a conformally rescaled metric, lowering an index multiplies the component by $e^{2f}$ and raising an index multiplies by $e^{-2f}$. These relationships hold for tensors of higher rank as well.

# 4 Derivation of the exponential equivalence

We now derive an explicit expression for the function $f(x, y, z)$ that connects the index transformation with the product of exponential factors $\exp(-x/y)$ and $\exp(z)$. Consider three algebraic variables $x, y, z \in \mathbb{R}$, with $y \neq 0$. Define

$$f(x, y, z) = z - \frac{x}{y}. \qquad (8)$$

Using the exponential identity (3), we have

$$e^{2f(x,y,z)} = \exp\left(2z - \frac{2x}{y}\right) = \exp(2z) \exp\left(-\frac{2x}{y}\right) = \left(\exp(z)\right)^2 \left(\exp(-x/y)\right)^2, \qquad (9)$$

where the exponent $-2x/y$ has been written as $-\frac{x}{y}$ twice for clarity. We choose the base metric $\gamma_{ij} = \delta_{ij}$, the Kronecker delta. Substituting $e^{2f}$ into the conformal metric (4), we obtain

$$g_{ij}(x, y, z) = \exp\left(2z - \frac{2x}{y}\right) \delta_{ij}. \qquad (10)$$

Lowering the index using (6) gives

$$v_i = g_{ij}v^j = \exp\left(2z - \frac{2x}{y}\right) \delta_{ij}v^j = \exp\left(2z - \frac{2x}{y}\right) v^i, \qquad (11)$$

where the last equality uses $\delta_{ij}v^j = v^i$. Taking the square root of the exponential factor yields [4]

$$v_i = \exp(z - x/y) v^i = \exp(z) \exp\left(-\frac{x}{y}\right) v^i, \qquad (12)$$

which shows that the covariant component is obtained by multiplying the contravariant component by the product $\exp(z) \exp(-x/y)$. Equation (12) is the desired equivalence between the operation of index lowering and a multiplicative exponential factor. Conversely, raising the index using (7) gives

$$v^i = \exp(-z + x/y) v_i, \qquad (13)$$

demonstrating that the exponential factors cancel when a covariant index is converted back to a contravariant one. The identity $\exp(z - x/y) \exp(-z + x/y) = 1$ follows from the reciprocal property of the exponential function, ensuring consistency with the fundamental identity $g^{ij}g_{jk} = \delta^i_k$.
## 4.1 Physical and geometric interpretation

The appearance of exponential factors in the metric is reminiscent of conformal transformations in differential geometry, where a scalar field rescales the metric. In conformal field theory one often writes $\tilde{g}_{\mu\nu} = e^{2\phi}g_{\mu\nu}$, and the factor $e^{2\phi}$ controls how lengths are distorted by the rescaling. When such a conformal factor depends on coordinates, raising and lowering indices introduces corresponding powers of $e^{2\phi}$. Our construction chooses a specific scalar function $f(x, y, z) = z - x/y$ whose exponential splits into two factors: a decay factor $\exp(-x/y)$ and a growth factor $\exp(z)$. This decomposition illustrates that index transformations can be recast as a multiplicative weighting by exponential functions. The algebraic nature of $x, y, z$ does not limit the interpretation; for instance, in thermodynamics the Boltzmann factor $\exp(-E/kT)$ converts between energy and probability, and in differential geometry conformal factors are used to study curvature.

# 5 Conclusion

We have demonstrated that the process of converting a contravariant vector component to a covariant component via the metric can be expressed in terms of an exponential scale factor. By introducing a scalar function $f(x, y, z) = z - x/y$, whose exponential satisfies $e^f = e^z e^{-x/y}$, and choosing a conformally scaled metric $g_{ij} = e^{2f}\delta_{ij}$, we derived the relation $v_i = e^z e^{-x/y} v^i$. The derivation relied only on the basic definitions of raising and lowering indices and on the functional equation of the exponential function. This explicit correspondence clarifies how exponential factors can serve as intermediate objects in tensor transformations, bridging index notation and conformal geometry. Our proof highlights the power of the exponential function in expressing multiplicative scaling and may prove useful in contexts ranging from general relativity to statistical mechanics, where exponential weights naturally arise.

## References

[1] Munson, Todd. "Mesh shape-quality optimization using the inverse mean-ratio metric." Mathematical Programming 110.3 (2007): 561-590.

[2] Ahmed, Faizan. "Tensor Calculus in General Relativity: A Bridge Between Mathematics and Theoretical Physics." Frontiers in Applied Physics and Mathematics 1.3 (2024): 217-250.

[3] "Exponential function," Wikipedia. The article explains that the exponential converts sums into products, i.e. $\exp(x + y) = \exp(x)\exp(y)$, and that $\exp(-x) = 1/\exp(x)$.

[4] Sochi, Taha. Tensor Calculus Made Simple. Taha Sochi, 2022.