# Christoffel's coefficients under laplace's formalism

Olivier Maurice

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Olivier Maurice. Christoffel's coefficients under laplace's formalism. 2024. hal-04476685

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Distributed under a Creative Commons Attribution 4.0 International License
# CHRISTOFFEL'S COEFFICIENTS UNDER LAPLACE'S
FORMALISM

Olivier MAURICE, AFSCET

2024, February 18th

## 1. INTRODUCTION

In [2] and [3] we develop various proposals for the tensor analysis of networks under Laplace's formalism. Between many equations we submit replacing time derivative by the simple product with $k$, Laplace's operator. This may astonish some readers and call for some more details.

## 2. BASIC WORKING LEADING TO CHRISTOFFEL'S COEFFICIENTS

We consider a vector $\vec{l}$ defined in the edge space on the base $e_\alpha$, itself defined by the fiber $u_k[2]$. A little differenciation on $\vec{a}$ can be written:

$$ (1) \qquad d\vec{l} = \vec{e}_\alpha dl^\alpha + l^\alpha d\vec{e}_\alpha $$

The $u_k$ defining a parametrized hypersurface of parameter $t$, we can write:

$$ (2) \qquad \frac{d\vec{l}}{dt} = \vec{e}_\alpha \frac{dl^\alpha}{dt} + \frac{d\vec{e}_\alpha}{dt} l^\alpha $$

But we can write differently the second term. It's a very important step because often we don't know the time derivative of the base vector, to see it can have no meaning, while their derivation depending on the space elements has a strong meaning. So, as usually made we write:

$$ (3) \qquad \frac{d\vec{e}_\alpha}{dt} = \frac{\partial \vec{e}_\alpha}{\partial q^k} \frac{dq^k}{dt} $$

Then we use:

$$ (4) \qquad \frac{\partial \vec{e}_\alpha}{\partial q^k} = \vec{e}_\sigma \Gamma^\sigma_{k\alpha} $$

$\Gamma^\sigma_{k\alpha}$ is a first kind Christoffel's coefficient. Replacing this expression in 2 gives:

$$ (5) \qquad \frac{d\vec{l}}{dt} = \vec{e}_\alpha \frac{dl^\alpha}{dt} + \frac{\partial \vec{e}_\alpha}{\partial q^k} \frac{dq^k}{dt} l^\alpha = \vec{e}_\alpha \frac{dl^\alpha}{dt} + \vec{e}_\sigma \Gamma^\sigma_{k\alpha} \frac{dq^k}{dt} l^\alpha $$

and so:

$$ (6) \qquad \frac{d\vec{l}}{dt} = \vec{e}_\alpha \left( \frac{dl^\alpha}{dt} + \Gamma^\alpha_{k\sigma} \frac{dq^k}{dt} l^\sigma \right) $$
which leads under Laplace's formalism [1] to:

$$ (7) \qquad k\vec{l} = \vec{e}_\alpha (kl^\alpha + \Gamma^\alpha_{\nu\sigma} kq^\nu l^\sigma) $$

This have meanings because $l$ as $q$ are respectively currents and loads in the edge space. As they are linked with electromagnetic fields, they can be written in the Laplace's domain.

### 3. CHRISTOFFEL'S SYMBOLS IN A SPACE TRANSFORMATION

Going from the edge space to the cycle one we obtain expressions of the form:

$$ (8) \qquad E^\alpha = \frac{d}{dt} (g^{ab} L_{ba} g^{a\mu}) a_\mu $$

here $a_\mu$ is a cycle current, $L$ the electromagnetic inertia and $g$ the metric of the edge space or its inverse. We can define a cycle load deducted from the cycle current:

$$ (9) \qquad Q_\mu = k^{-1} a_\mu $$

Let's write:

$$ (10) \qquad \begin{aligned} \frac{d}{dt} (g^{\alpha b} L_{ba} g^{a \mu}) a_{\mu} &= \frac{d}{dt} g^{\alpha b} (L_{ba} g^{a \mu}) a_{\mu} + g^{\alpha b} \left(\frac{d}{dt} L_{ba}\right) g^{a \mu} a_{\mu} + \dots \\ & \quad + g^{\alpha b} L_{ba} \left(\frac{d}{dt} g^{a \mu}\right) a_{\mu} + \left(g^{\alpha b} L_{ba} g^{a \mu}\right) \frac{d}{dt} a_{\mu} \end{aligned} $$

Knowing that $dL/dt = 0$ we obtain:

$$ (11) \qquad \frac{\partial g^{\alpha b}}{\partial Q_{\nu}} \frac{d Q_{\nu}}{dt} (L_{ba} g^{a \mu}) a_{\mu} + g^{\alpha b} L_{ba} \frac{\partial g^{a \mu}}{\partial Q_{\nu}} \frac{d Q_{\nu}}{dt} a_{\mu} + L^{\alpha \mu} \frac{d}{dt} a_{\mu} $$

We see here the great advantage to use the proposal of considering $g$ as a metric [3]. Now using:

$$ (12) \qquad \Gamma^{\alpha b, \nu} = \frac{\partial g^{\alpha b}}{\partial Q_{\nu}} $$

and $(L_{ba}g^{a\mu}) a_\mu = \Phi_b$ this leads to:

$$ (13) \qquad \frac{d}{dt} (g^{\alpha b} L_{ba} g^{a \mu}) a_{\mu} = \Gamma^{\alpha b, \nu} a_{\nu} \Phi_{b} + g^{\alpha b} L_{ba} \Gamma^{a \mu, \nu} a_{\nu} a_{\mu} + L^{\alpha \mu} \frac{d}{dt} a_{\mu} $$

We can create a new object $\Omega$ with:

$$ (14) \qquad \Omega^{\alpha \mu, \nu} = g^{\alpha b} L_{ba} \Gamma^{a \mu, \nu} $$

And using Laplace's formalism we obtain finally:

$$ (15) \qquad E^{\alpha} = \Gamma^{\alpha b, \nu} a_{\nu} \Phi_{b} + \Omega^{\alpha \mu, \nu} a_{\nu} a_{\mu} + k L^{\alpha \mu} a_{\mu} $$
## 4. ON THE LEGITIMACY OF DISTRIBUTIVITY APPLIED TO LAPLACE'S OPERATOR

The unique variables being expressed as sinusoidal functions are the currents and loads. When the time derivative is applied on the metric or an impedance, it cannot be by the fact replaced by a product with Laplace's operator. Fortunately, this time derivatives can generally be replaced by a space derivative multiplied by the time derivative of the space, using as previously for an object O:

$$ (16) \qquad \frac{d}{dt}O = \frac{\partial O}{\partial Q^{\sigma}} \frac{d}{dt}Q^{\sigma} \rightarrow \frac{\partial O}{\partial Q^{\sigma}} kQ^{\sigma} $$

That's through this mechanism that we can still use the classical replacement of the time derivative even when at first, the object is not an exponential function.

## 5. ON THE TIME MEANING

The simplest image for thinking of time is probably a clock, turning infinitely. A complete revolution of its clock hand can be assimilated with a time duration or period T. Once this period defined, things become easy. Laplace's fundamental spectrum is developed on all the period multiples k/T. And the particles energy E is linked with this period also: E = -iħk. Here we see that Laplace's operator is as legitimate as the classical time for being the fourth axes of space.

The concept of time was drastically reviewed after Einstein's relativity formulation. Lorentz-Poincaré's transformation changed Galilée's one adding principally the term $γ = \sqrt{1 - v^2/c^2}^{-1}$. First Galilée's law gives $x' = x + Vt$. By analogy, the time follows something similar with its equivalent length ct:

$$ (17) \qquad ct' = ct + Vτ $$

The time length $τ$ translating the information displacement $x/c$. This gives: $t' = t + Vx/c^2$. Adding the relativistic correction we arrive to:

$$ (18) \qquad t' = \gamma \left( t + \frac{Vx}{c^2} \right) $$

Dividing both members by the time in the referential in movement, we write the ratio of time duration in both referentials:

$$ (19) \qquad \frac{t'}{t} = \gamma \left( 1 + \frac{ax}{c^2} \right) $$

$a$ is this time the acceleration in the observed referential. But the acceleration is equal to the gravitation acceleration $g$, due to the equivalence between the inertia mass and the weighted one. The gravitational field itself derives from the gravitation potential $Φ$: $g = Φ/x$, leading to[4]:

$$ (20) \qquad \frac{t'}{t} = \gamma \left( 1 + \frac{\Phi}{c^2} \right) $$
This relation can be applied to periods with:

$$ (21) \qquad T' = T\gamma\left(1 + \frac{\Phi}{c^2}\right) $$

and so, Laplace's transform becomes:

$$ (22) \qquad \frac{k}{T'} = \frac{k}{T} \left[ \gamma \left( 1 + \frac{\Phi}{c^2} \right) \right]^{-1} $$

In presence of a gravitational field, the period increases and the Laplace's fundamental component decreases: the energy spectrum is displaced on the left. Basically this means that one clock turn represents a higher time duration $T'$ rather than without gravitation $T$. Near black holes, this period may increase until a value for which the time disappears, i.e. that the period becoming so long that the time perception cannot discerns the clock turning as if they was no longer any time evolving[5]. Seen from Laplace's operator, all work as if there was only one frequency, all the spectrum being jailed in this single spectrum ray of very straight width. And the frequency value of the spectrum ray tends to zero.

Perhaps the best representation of the time goes through this duality with Laplace's operator. The clock rotation describes both the time rhythm and the fundamental energy spectrum frequency.

## 6. CONCLUSION

It was not so obvious to accept the use of Laplace's transform applied for differential geometry. In fact it has meaning that this use being possible. It recalls that in tensor analysis of networks, the fundamental base is a harmonic function intrinsically described by Laplace's formalism.

Following quantum mechanics, wave vector is a harmonic function. This context is favorable to the application of the relationship discussed here. But this is also a good argument for justifying its use.

Equation 7 resume our discussion showing the covariant derivative under Laplace's formalism. But it must be kept in mind that the strong condition for being authorized to do it is the harmonic behavior of the variables concerned.

The harmonic representation associated with Laplace's transform gives as many information as the classical time rhythm do. Both variables are dual: it's all the more true when we think of particles description.

## REFERENCES

[1] Murray R.Spiegel, "Transformées de Laplace". Schaum series. Mc Graw Hill edition.

[2] Olivier Maurice. From $\mathbb{R}^4$ curvilinear space to the cycle space. 2024. (hal-04448080)

[3] Olivier Maurice. Inclusion of the tensor analysis of networks into differential geometry. 2023. (hal-04286165)

[4] Einstein, A. (1989). Oeuvres choisies. Éditions du Seuil.

[5] Stephen Hawking, "L'univers dans une coquille de noix" or "The Universe in a Nutschell". Odile Jacob edition. 2001;