Mass as worldline tension?

Question

In Special Relativity, the equation of motion of a particle of proper mass $m$ is \begin{equation}\tag{1} \frac{d p^a}{d \tau} = \mathfrak{F}^a, \end{equation} where $p^a = m \, u^a$ is the 4-momentum, $\tau$ is the particle's proper time along its worldline in spacetime and $\mathfrak{F}^a$ is the 4-force acting on the particle. A free particle ($\mathfrak{F}^a = 0$) has a straight worldline in spacetime, while a force would curve it (as defined in an inertial frame).

I have read somewhere (can't find back the paper) that proper mass ($m$ or $m c^2$) could be interpreted as the particle's worldline tension in spacetime. By bending or curving the worldline, a force would produce a tension on the worldline, which could be "seen" as a kind of material string stretching in spacetime. If this idea isn't a crackpot concept, then inertia could be interpreted as the difficulty in curving the "material" worldline.

Maybe this is a boundary issue? In the very far past and future, an "external" agent is pulling the straight worldline, adding a tension to it. Thus the worldline would resists any other force applied to it.

How this idea could be made more precise, mathematically? How could we define the worldline tension, especially since it doesn't have clear end points (the worldline is extending to past and future infinities in spacetime)?

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EDIT: I'm tempted to write (1) as the following: \begin{equation}\tag{2} \mathfrak{F}^a - m \, \frac{d u^a}{d \tau} = 0, \end{equation} and interpreting the last term on the left side as an "inertial force", i.e. the "external" tension applied on the worldline by an "external" agent. The right part (i.e. = 0) is a way of stating that the worldline is always in an equilibrium state (obviously, a worldline isn't moving in spacetime itself). But I feel that this interpretation is pretty sterile and arbitrary, and have a metaphysical feel.

This idea is a bit like writing the Einstein equation (in General Relativity) as \begin{equation}\tag{3} T_{\mu \nu} - \frac{1}{8 \pi G} \, G_{\mu \nu} = 0, \end{equation} and interpreting the last term as the energy-momentum of the gravitational field itself, while the right part could be interpreted as total energy-momentum (0 for any spacetime). This is pretty arbitrary.

Answer 1

I haven't read the contents of these references.
So, I can't comment on them.
I have merely tracked down what appear to be relevant references on the topic....
possibly what the OP "read somewhere".

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Olivier Costa de Beauregard's
"Isomorphisme de la dynamique relativiste des systèmes de points et de la statique classique des systèmes de fils" - Cahiers de physique, 80, avril 1957, pp. 137-148
http://www.costa-de-beauregard.com/fr/wp-content/uploads/2012/09/OCB-1957-10.pdf

Here is the abstract:

Sommaire: On met en evidence l'isomorphisme entre la dynamique relativiste du point et la statique classique du fil (voir tableau de correspondances 1) entre la dynamique relativiste des systèmes de charges electrisees de Wheeler-Feynman [11] et la statique classique des systemes de fils en interaction. Accessoirement, l'on discute de la définition covariante relativiste du barycentre [9, 12] et de l'énoncé relativiste des théorèmes généraux de la dynamique [12] puis de la relation entre l’irréversibilité macroscopique du rayonnement et celle de la thermodynamique.

which Google Translate try to translate as

"Isomorphism of relativistic dynamics of point systems and classical static of wire systems"

Summary: The isomorphism between the relativistic momentum of the point and the classical static of the yarn (see correlation table 1) between the relativistic dynamics of Wheeler-Feynman's electrified charge systems [11] and the classical static of the systems is demonstrated. son in interaction. Incidentally, we discuss the relativistic covariant definition of the centroid [9, 12] and the relativistic statement of the general theorems of dynamics [12] and then the relation between the macroscopic irreversibility of radiation and that of thermodynamics.

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I found that reference from this article at
https://arxiv.org/abs/1711.03568
Calin Galeriu's
"Relativistic Point Particles and Classical Elastic Strings"
whose abstract reads

We extend the previous work of Olivier Costa de Beauregard regarding the isomorphism between the equation describing the motion of relativistic point particles and the equation describing the static equilibrium of classical elastic strings, by comparing the Lagrangians of these two systems.

Answer 2

What is important is not the fact that equation (1) describing the change in the 4-momentum can also be looked upon as describing the change in the worldline tension, what is important is the replacement of the material point particle model with an infinitesimal length element model. The 4-force in equation (1) acts on a point particle, but in the worldline string model we have a linear 4-force density acting on an infinitesimal worldline segment. The material point interacts with other material points on its light cone, and this shows up as Dirac delta functions in the equations. The infinitesimal worldline segment interacts with other infinitesimal segments in a very different way. Because of the geometry of the corresponding length elements, one can recover the expression of the electromagnetic interaction, up to a factor.

https://arxiv.org/abs/1712.02213

Electric charge in hyperbolic motion: arcane geometrical aspects