The Twin Paradox in Relativistic Physics: Analyzing the Limit as α Approaches Infinity and t' Approaches Zero

Let's explore a claim related to the twin paradox in relativistic physics. Specifically, we investigate the equation:

t frac{c}{alpha} sinhleft(frac{alpha t}{c}right)

where α represents the constant proper acceleration in the accelerated frame of the twin paradox, and t is the coordinate time in an inertial reference frame. Here, we will analyze the behavior of this equation as t' to 0 and α to infty.

Introduction to the Twin Paradox and Rindler Coordinates

The twin paradox is a thought experiment that examines the effects of special relativity on time dilation. When two twins start together and one undergoes constant acceleration, they will experience different elapsed times when they reunite.

Rindler coordinates describe the accelerated frame of reference. The transformation between these coordinates and an inertial frame is complex, and the given equation aims to capture how the proper time experienced by the accelerated twin can be expressed in terms of coordinate time in an inertial frame.

Analysis of the Equation

To analyze the behavior of t frac{c}{alpha} sinhleft(frac{alpha t}{c}right) as t' to 0 and α to infty, we need to understand the Taylor expansion of the hyperbolic sine function for small arguments.

Step 1: Expansion of the Hyperbolic Sine for Small Arguments

For small x, the Taylor expansion of the hyperbolic sine function is:

(sinh(x) approx x - frac{x^3}{6} mathcal{O}(x^5))

Let (x frac{alpha t}{c})

Substituting this into the hyperbolic sine:

(sinhleft(frac{alpha t}{c}right) approx frac{alpha t}{c} - frac{1}{6}left(frac{alpha t}{c}right)^3 mathcal{O}left(left(frac{alpha t}{c}right)^5right))

Step 2: Substituting Back into t

Substitute this expansion into the equation for t:

(t frac{c}{alpha} sinhleft(frac{alpha t}{c}right) approx frac{c}{alpha} left(frac{alpha t}{c} - frac{1}{6}left(frac{alpha t}{c}right)^3 mathcal{O}left(left(frac{alpha t}{c}right)^5right)right))

Simplifying the expression:

(t approx t - frac{1}{6}frac{alpha^2 t^3}{c^2} mathcal{O}left(frac{alpha^4 t^5}{c^4}right))

Step 3: Limits

As (t to 0):

When (t) approaches zero, all higher-order terms in the expansion vanish, leaving:

(t to t)

This result is consistent with the fact that at (t 0), the inertial and accelerated frames coincide.

As (α to infty):

Let (t) be fixed and consider the term (frac{1}{6}frac{alpha^2 t^3}{c^2}). As (α to infty), this term diverges. Therefore, (t to infty) for any fixed (t > 0).

Combined Limit (t to 0) and (α to infty):

If both (t to 0) and (α to infty) are considered simultaneously, the behavior depends on the rate at which (t) approaches zero relative to (α).

For (αt to 0), then (t to t) as higher-order terms vanish. For (αt) held constant or growing, then (t to infty).

The behavior of (t frac{c}{α} sinhleft(frac{αt}{c}right)) critically depends on the relationship between (t) and (α). For (t to 0), inertial and accelerated times converge to (t). However, for (α to infty), (t) can diverge unless (t) is correspondingly small enough to suppress higher-order contributions.