Using relativistic kinematics to model a chapter in "Einstein's Dreams"

Background

I'm a MSc in financial maths student and LOVE physics. I wanted to model this chapter in Alan Lightman's novella.

14 MAY 1905

There is a place where time stands still. Raindrops hang motionless in air. Pendulums of clocks float mid-swing. Dogs raise their muzzles in silent howls. Pedestrians are frozen on the dusty streets, their legs cocked as if held by strings. The aromas of dates, mangoes, coriander, cumin are suspended in space.

As a traveler approaches this place from any direction, he moves more and more slowly. His heartbeats grow farther apart, his breathing slackens, his temperature drops, his thoughts diminish, until he reaches dead centre and stops. For this is the centre of time.

From this place, time travels outward in concentric circles—at rest at the centre, slowly picking up speed at greater diameters.

Who would make pilgrimage to the center of time? Parents with children, and lovers.

(full text here)

My Attempt

Let's use

$\Delta \tau = \int_{t_1}^{t_2} \left( 1 - e^{-\alpha r(t)} \right) dt$

where

• $\Delta \tau$: the dilated time (often used in general or special relativity and quantum coherence decay models).

• $(t_{1})$ to $(t_{2})$: the integration bounds representing the beginning and end of the time interval.

• $\alpha$: a specific coupling constant (or, attenuation coefficient, or rate parameter?)

• $r(t)$: a time-varying function or spatial radius, representing the distance, energy profile, or physical separation that modifies the rate.

When a traveler leaves the centre and walks back to the outer world, they experience an absolute time mismatch. We can calculate the total time lost by integrating the time differential along their escape path over a path interval from time $t_1$ to $t_2$.

Because the integrand $\left( 1 - e^{-\alpha r(t)} \right)$ is bounded by $0$ and $1$, the time experienced by the traveler ($\Delta \tau$) will always be strictly less than the time that elapsed for their friends in the outer world ($\Delta t = t_2 - t_1$).

$\Delta \tau \ll \Delta t$

This stark inequality explains the mathematical tragedy at the end of the text: when they finally step out of the town's field, they realize that while their embrace felt like it lasted mere seconds, centuries have ticked away, leaving them stranded in a future world where everyone they once knew is long gone.

Let's run a random numerical simulation for three different couples who decide to share a $10$-minute embrace (from their own perspective) at various distances from the centre of time.We will use our analytical metric

$d\tau = \left(1 - e^{-\alpha r}\right)dt$

and set the town's field sensitivity parameter to $\alpha = 0.5$. If each couple experiences exactly

$\Delta \tau = 10 \text{ minutes}\ (0.1667 \text{ hours})$

we can calculate exactly how much time passes for the rest of the world $\Delta t$ using:$\Delta t = \frac{\Delta \tau}{1 - e^{-\alpha r}}$

• Couple A: Standing in the Outer Suburbs ($r = 10 \text{ km}$). They are far from the centre, where the field is weak.\begin{align}\Delta t &= \frac{0.1667}{1 - e^{-0.5 \times 10}}\ &= \frac{0.1667}{1 - e^{-5}}\ &= \frac{0.1667}{1 - 0.0067}\ &= 0.1678\ \text{ hours.}\end{align} For their $10$-minute hug, $10$ minutes and $4$ seconds pass for the outside world. Time is almost perfectly synced.

• Couple B: Standing in the Inner Ring ($r = 0.1 \text{ km} = 100 \text{ metres}$). They have walked deep into the historic town, close to the temporal core.\begin{align}\Delta t &= \frac{0.1667}{1 - e^{-0.5 \times 0.1}}\ &= \frac{0.1667}{1 - e^{-0.05}}\ &\approx \frac{0.1667}{1 - 0.9512}\ &= \frac{0.1667}{0.0488}\ &= 3.416 \text{ hours.}\end{align} They lean against a building for what feels like a quick $10$-minute romantic embrace. When they open their eyes, $3$ hours and $25$ minutes have flashed by in the outer world. The sun has completely changed position in the sky.

• Couple C: Standing at the Absolute Threshold ($r = 0.00001 \text{ km} = 1 \text{ centimetre}$). They are standing practically at dead centre, right where the "statues" are frozen.\begin{align}\Delta t &= \frac{0.1667}{1 - e^{-0.5 \times 0.00001}}\ &\approx \frac{0.1667}{0.000005}\ &= 33,340 \text{ hours.}\end{align} They hold each other for what feels like $10$ brief minutes. They step backward, walk out of the town, and realize that $3.8$ years have passed for the rest of the planet. Their friends have moved away, their leases have expired, and the outer world has completely left them behind.

My Question

Would this make any sense? Where did I go wrong?