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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)?

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{2} 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.

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)?

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.

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)?

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)?

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{2} 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.

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.

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)?

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)?