Twin Paradox and Proper Time

Professors Mohazzbi and Luo [1] published “Despite several attempts have been made to explain the twin paradox … none of the explanations … resolved the paradox. If the paradox can be ever resolved, it requires a much deeper understanding … of the theory of relativity”. The deeper understanding of re-solving the paradox is by applying more explicit definitions of proper time interval


Introduction
Professors Mohazzbi and Luo documented the failures to resolve the twin paradox [1], to their satisfaction.The following resolves their concerns.Explicit definitions of Lorentz transform, time dilation, proper time and aging time will be given and applied to the twin paradox analysis.The transient/dynamic nature of the Lorentz transform will be introduced.The analysis demonstrates the astronaut returning to Earth ages the same as his twin that stayed on Earth, using constant velocity reference frames in special theory.When the analysis is done using constant acceleration reference frames in general relativity, the result is the same.
Δt n and Δx n are frame n coordinates where the velocity from frame 0 to frame n is 0n n n v x t = ∆ ∆ (There are many v 0n values representing the same Δτ value.)(5) When Δx 0 = 0 and Δt 0 , frame 0 has absolute velocity zero.By inspection of (4), a frame at rest must have its Δx = 0. Another way to tell if a frame is at rest in space, is an object in the frame has no kinetic energy, it is at rest mass.Δt 0 (Δτ equals Δt 0 ) is an invariant of the Lorentz transform.A given a value of Δτ g restricts the transformed coordinates allowed in a transformed Lorentz reference frame.When coordinates are transformed from frame A to frame B using any constant velocity between the frames, the coordinates (Δt A and Δx A ) and coordinates (Δt B and Δt B ) are restricted by the Minkowski metric.
Proper time interval is defined as the Δt 0 (clock time interval between two events) observed in your frame (0) when the two events happen in frame (0).If the two events did not happen in your frame (0), Δt 0 can be established in your frame (0), Event 1 flashes a green light, Event 2 flashed a red light, and the distances to the two events is known.If the light sources are moving there will be a frequency shift of the red and green flashes.
Consideration of the transit/dynamic nature of the Lorentz transform: consider two rest frames A (Δx A equals 0) and B (Δx B equals 0) that are Δx D distance from frame A to frame B. Let a clock move from frame A to frame B at a constant velocity v (Event 1), stops (Event 2), and return to frame A at a velocity-v (Event 3).Upon return to frame A, it stops (Event 4).
Start 0  Consider the relationship between proper time interval Δτ and aging interval.
An example of an aging interval is the time between the birth (Event 1) and death of a day fly (Event 2), which is represented by Δτ = 24 hours, which is the life span of a day fly.Proper time interval (Δτ) can represent the aging interval, which is an interval between two events.This all means astronaut returning to Earth ages the same as his twin that stayed on Earth.

Characteristics of General Relativity Schwarzschild Metric [2] [3] [4]
The Schwarzschild Metric models an astronaut accelerating away from Earth and then revering his acceleration, returning to Earth.It is assumed the acceleration and velocity in no way affects his heath.The version of the metric is: n identifies the reference frame.1, 2, 3, n =  .Δt n is time interval in frame n.Δx n is length interval in frame n. α F is parameter representing the constant gravity force moving an object Δx n distance in Δt n time.α F can have different values in a different frame.
This illustrates, snap shots of coordinates as the clock moves from one stationary Lorentz reference to another and then returns.The key considerations are Lorentz frames, standing still in space with their Δx equal to zero.The going out Δτ equals the return Δτ.The proper time interval Δτ (between Start and Event 4) is 2Δx D /v.Time dilation of an interval is observable only for an instant.

(α = 1 ,
represents no force and the Schwarzschild metric becomes the Minkowski metric.)(α = 0, represents infinite force.)Δτ, proper time interval, is invariant of Δt k and Δx k coordinates in frame (k) where α F can have different values in different frames representing a different gravity force in that frame.Consider astronaut rockets from Earth at constant acceleration α A for a proper time interval of 1/4 (Δτ T ) and reveres his acceleration for 1/4 (Δτ T ) returning to zero velocity.Then keeping acceleration toward Earth for 1/4(Δτ T ) and again revering his acceleration away from Earth for 1/4(Δτ T ), returning to Earth with zero velocity.The total proper time interval (Δτ T ), experienced by the astronaut is Δτ T .His twin has stayed on Earth for the same proper time interval (Δτ T ).Both twins will have aged at the same proper time interval.D. Lem DOI: 10.4236/jamp.2024.121002