Twin Paradox

The Twin Paradox is a classic thought experiment in special relativity that explores how time passes differently for two observers following different paths through spacetime. While the standard formulation involves a sudden turnaround, a more physically rigorous approach involves continuous acceleration.

In the scenario analyzed below, we assume the Resting Twin remains stationary in an inertial frame. The Travelling Twin begins the experiment with an initial high velocity moving to the left of the resting twin. However, throughout the entire journey, the traveller engages their engines to maintain a constant proper acceleration directed to the right.

This "rightward" acceleration acts initially as a braking force. It gradually slows the "leftward" moving traveller until they come to a momentary halt at a maximum distance. The continued acceleration then propels them back toward the Resting Twin. This creates a smooth, hyperbolic worldline rather than a jagged path.

The core of the paradox lies in the comparison of clocks. Naive intuition suggests that because motion is relative, each twin should see the other's clock running slow (time dilation), leading to a contradiction upon reunion. However, as the following breakdown demonstrates, the situation is not symmetric: the Resting Twin occupies an inertial frame, while the Travelling Twin occupies a non-inertial (accelerated) frame, leading to very different mathematical descriptions of the passage of time.

1. In the Resting Twin's Frame (Inertial)

Metric: Spacetime is described using the standard Minkowski metric $η_{μ ν}$ , which is characteristic of the flat geometry found in special relativity.
Geometry: In this frame, the worldline of the Earth twin is a straight line, representing a geodesic in spacetime. Conversely, the worldline of the travelling twin is curved due to acceleration, making it non-geodesic.
Result: Within the geometry of a Lorentzian manifold, a "straight" timelike path between two points (events) actually maximises the proper time. This is the "longest" possible path in terms of duration; therefore, any deviation from this geodesic path, such as the traveller's journey, necessarily results in a shorter elapsed proper time.

2. In the Travelling Twin's Frame (Non-Inertial)

Metric: When the travelling twin accelerates, their frame of reference becomes non-inertial. To describe this, coordinates must become curvilinear—specifically Rindler coordinates during the phase of constant proper acceleration. Consequently, the metric tensor $g_{μ ν}$ is no longer a constant diagonal matrix.
The line element in this frame typically takes the form $d s^{2} = (1 + a x)^{2} d t^{2} - d x^{2}$ , where $a$ is the acceleration and $x$ is the distance from a specific reference point.
The "Gravity" Term: The term $(1 + a x)^{2}$ represents a "gravitational" potential that is explicitly dependent on the distance $x$ . According to the equivalence principle often utilised in general relativity, the effects of acceleration are locally indistinguishable from a gravitational field.
Calculation:
- Traveler: Remains at $x = 0$ in their own frame, meaning $(1 + a x) = 1$ ; their clock ticks at a normal rate relative to their position.
- Earth: From the traveller's perspective, the Earth is at a vast distance $x$ . Because $x$ is extremely large, the $g_{00}$ component (the $(1 + a x)^{2}$ term) becomes enormous.
Result: The total proper time $τ$ for the Earth is found by integrating the metric along its worldline: $τ = \int \sqrt{g_{μ ν} {\dot{x}}^{μ} {\dot{x}}^{ν}} d λ$ .
- During the traveller's acceleration/turnaround phase, the huge magnitude of the "gravitational" potential term dominates this integral.
- This "pumps" a massive amount of proper time into the Earth's total duration.
- Even though the Earth is moving relative to the traveller (which would normally induce time dilation and slow the Earth's clock), this gravitational potential effect significantly overwhelms the motion-based slowing, ensuring the Earth twin is older upon reunion.