Twin paradox - a logical explanation

  Note: This article summarises the ideas presented in: "Vlad, I. (2026) Time vs. Duration: A Reinterpretation of Special Relativity", a Vixra preprint:

Time vs. Duration: A Reinterpretation of Special Relativity

  1. Introduction

The twins' paradox is commonly explained as a consequence of time dilation and the relativity of simultaneity. However, the standard interpretation often relies on simplified Minkowski diagrams and ambiguous coordinate assignments, which can obscure the physical meaning of the turnaround event. In particular, the commonly described "jump" in simultaneity is frequently introduced as an explanatory device rather than as a result derived directly from the mathematics.

Consider the twins' paradox for a traveller moving at a velocity of v = 0.8c, with a turnaround distance of 4 light-years. This gives a coordinate time interval of $\Delta t_{0}=5$ years and a proper time interval of τ = 3 years. The corresponding spacetime diagram is shown in Figure 1.

Figure 1. Conventional schematic Minkowski representation of the twin paradox for v = 0.8c

In the conventional diagram, the green dotted line of simultaneity passing through the turnaround point intersects the ct axis at 1.8 years instead of 3 years. This occurs because Minkowski diagrams are usually drawn schematically rather than to scale. A Lorentz-consistent Minkowski diagram is shown in Figure 2.

Figure 2. Lorentz-consistent Minkowski diagram illustrating the correct assignment of turnaround coordinates and simultaneity structure for v = 0.8c

The construction of this diagram differs from the conventional schematic representation in two important ways.

First, the spacing of the pink grid representing the moving frame S' is not uniform in Euclidean space. Instead, the grid must satisfy the Minkowski geometry. Equal increments in the moving coordinates are therefore defined by:

• Equal increments in ct′ correspond to: ct − vx = constant increments

$ct=vx+{\frac {k}{\gamma }}$

• Equal increments in x′ correspond to: x − vct = constant increments

$x=vct+{\frac {k}{\gamma }}$

Second, the spacetime events must be assigned coordinates consistently.

T ($x=4,ct=5$) is the turnaround event expressed in the Earth frame S. It is not defined using the coordinates of S'. T' ($x'=4,ct'=5$) is the same turnaround event expressed in the traveller's frame S'. This point should not be confused with the event at ($x'=2.4,ct'=3$). P is the projection of T' onto the ct axis of the Earth frame. P' is the projection of T onto the traveller's frame S'.

These points all represent the same physical event, but each is expressed in a different coordinate system. Once the Lorentz transformation has been used to construct the S' grid (shown in pink), the coordinates assigned to the event in S must remain unchanged. Applying the Lorentz transformation again to those coordinates would amount to transforming the same event twice.

This possible misunderstanding can be illustrated using the S' simultaneity line passing through T. If interpreted incorrectly, it might suggest that the event has coordinates ($x'=0,ct'=3$) in S'. Projecting this point back onto the ct axis would then appear to give ($x=0,ct=1.8$), as indicated by the dotted green line in Figure 1.

According to this interpretation, however, a simultaneity line in S' cannot be drawn through T because T is already defined using the coordinates of the Earth frame S (Figure 2). The event cannot be reassigned different coordinates within the same construction. Instead, the original coordinates ($x=4,ct=5$) are retained while the event is represented within the S' grid as T' ($x'=4,ct'=5$). This explains why T' does not have the coordinates ($x'=2.4,ct'=3$).

Projecting T' onto the ct axis gives P ($x=0,ct=3$), which corresponds to the traveller's proper time during the outward journey (3 years), consistent with the predictions of Special Relativity.

To analyse the same event from the traveller's frame, it must first be expressed in the coordinates of S'. When projected onto the S' axes, the turnaround event (T) becomes P' ($x'=0,ct'=3$). Consequently, drawing another simultaneity line through T/P' is not meaningful because both labels already describe the same spacetime event. The same reasoning applies to the return journey.

This construction clarifies the meaning of simultaneity and the role of quantities that remain operationally invariant in Special Relativity. Although no inertial frame is fundamentally preferred, meaningful comparisons between observers require all quantities to be expressed in a common reference frame. In practice, this is simply a matter of expressing all measurements in the same coordinate system, much as a length can be expressed in either inches or centimetres before it is compared.

The chosen frame therefore serves only as a common basis for comparison; it does not become physically privileged.

It is also important to distinguish simultaneity from visual appearance. In Figure 2, the same spacetime event is assigned same coordinates in different reference frames ($ct_{T}=ct'_{T'}=5$), while the event itself remains unchanged. At first, the projections P and P′ appear to represent different coordinates because the Lorentz transformation has not yet been applied to relate them.

Because the Lorentz transformation is already built into the geometry of the moving frame, the coordinate relationships remain consistent, and the physical identity of the spacetime event is preserved, even though the local units of measurement differ between frames. Figure 2 is drawn from the perspective of S (the rest frame). To view the same geometry from the perspective of S' as the rest frame, the diagram must be sheared so that the ct′ axis becomes vertical while preserving the internal coordinate relationships. Applying this transformation produces Figure 3.

Figure 3. Lorentz-consistent Minkowski diagram illustrating the transformed simultaneity structure for v = 0.8c.

Since simultaneity is defined by time, every line perpendicular to the vertical time axis represents a plane of simultaneity. In this representation, P and P' lie on the same horizontal line, indicating that they correspond to the same spacetime event. The time coordinate of P is identical to the time coordinate of T' ($ct'_{T'}=5$). Because the Lorentz transformation is embedded in the S' grid but is not applied to the S grid, P appears on the ct axis with coordinate

$ct_{P}={\frac {ct'_{T'}}{\gamma }}=3$

(cyan dotted simultaneity line). A correction must therefore be applied to the ct coordinate. Initially, the projection of T' onto the S grid appears at ($x=2.4$, $ct=3$). These coordinates do not represent a single spacetime event in S. Instead, they arise from independent projections onto the x and ct axes and therefore cannot be combined into one event. A consistent event is recovered only after applying the appropriate Lorentz transformation.

Once this Lorentz-consistent correction is applied, the coordinates of P become ($x=0,ct=5$), while the projection of T' is located at ($x=4,ct=5$). This preserves the coordinate relationships between the two reference frames.

Furthermore, when viewed from the traveller's frame, the coordinate time associated with the ct axis appears dilated, with $ct'_{5}<ct_{5}$, while the spatial units in S' are contracted relative to those in S, as predicted by Special Relativity. This reflects the distinction between coordinate time and elapsed duration (proper time), which underlies the usual interpretation of "time dilation".

  2. Distinguishing Time from Duration

Much of the confusion surrounding the twins' paradox comes from treating two different concepts as though they were the same: coordinate time and proper duration.

In spacetime, coordinate time is one of the four coordinates describing an event, analogous to the spatial coordinates x, y, and z, although it remains physically distinct because of the spacetime metric.

Just as we do not directly measure x, y, or z, but rather the intervals Δx, Δy, and Δz, we do not measure time itself. Instead, we measure the interval Δt, which is the duration between two events.

The key distinction is:

Coordinate time provides the common temporal reference used to describe events. Proper duration depends on the geometry of an observer's world-line and therefore differs for observers following different inertial or accelerated paths.

This relationship is similar to the distinction between mass and weight. Mass remains the same regardless of the surrounding gravitational field, whereas weight depends on local conditions. Likewise, distinguishing coordinate time from proper duration can help clarify many of the common misunderstandings surrounding the twins' paradox without changing the mathematical framework of Special Relativity.

  3. Mathematical framework

Consider the following parameters:

Speed: v = 0.8c

Distance (Earth to the star, at rest): 4 light-years

Round-trip distance ($L_{0}$): 8 light-years

In the inertial frame (Earth):

$\Delta t_{0}={\frac {L_{0}}{v}}={\frac {8}{0.8}}$ = 10 years

The Lorentz factor is:

$\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-0.64}}}=1.67$

Because the traveller is moving relative to Earth, the measured distance is length-contracted:

$L'={\frac {L_{0}}{\gamma }}={\frac {8}{1.67}}=4.8{\text{ light-years}}$

The traveller's proper duration is therefore:

$\tau ={\frac {\Delta t_{0}}{\gamma }}={\frac {10}{1.67}}\approx 6{\text{ years}}$

The duration difference between the Earth's coordinate time and the traveller's proper duration is:

$D_{difference}=\Delta t_{0}-\tau =10-6=4{\text{ years}}$

  3.1. Velocity consistency check

From the traveller's perspective, the calculated velocity remains unchanged:

$v={\frac {L'}{\tau }}={\frac {4.8}{6}}=0.8c$

The same physical speed is obtained because both the measured distance and the measured duration change consistently under the Lorentz transformation.

  3.2. The return to the rest frame

The situation becomes more subtle when the spacecraft returns to Earth and is once again at rest relative to it. At this point, the distance between Earth and the star returns to its original value of 8 light-years (round trip).

Although the traveller's clock has recorded only 6 years of proper duration, this does not mean that time itself has been permanently altered. The traveller's clock measures the elapsed proper duration along the traveller's world-line; it does not measure the coordinate time of the Earth's reference frame.

Once the journey is complete and both twins are again at rest with respect to one another, physical quantities such as distance and coordinate time are once again defined within the same inertial frame. Under this interpretation, no paradox remains.

  3.3. Addressing common misconceptions

Incorrect calculation (after returning to the rest frame)

$v={\frac {L_{0}}{\tau }}={\frac {8}{6}}=1.33c$

This calculation incorrectly combines measurements taken in different reference frames. It uses the proper duration $\tau$, measured in the moving frame, together with the rest distance $L_{0}$, measured in the Earth's inertial frame.

The result is an apparent velocity greater than the speed of light, which contradicts the principles of Special Relativity.

Measurements made in different reference frames cannot be mixed directly. Each frame has its own internally consistent system of length and time measurements. When the spacecraft returns to the Earth's rest frame, the measured distance returns to its rest value. The corresponding time measurement must also be expressed in that same frame, since both quantities are related by the Lorentz transformation.

Correct calculation (after returning to the rest frame)

$v={\frac {L_{0}}{\tau \times \gamma }}={\frac {8}{6\times 1.67}}={\frac {8}{10}}=0.8c$

The expected velocity is recovered.

Likewise,

$\Delta t_{Spaceship}={\frac {\gamma \times L'}{v}}={\frac {L_{0}}{v}}={\frac {8}{0.8}}=10{\text{ years}}$

which agrees with the coordinate time measured in the Earth's frame.

  4. Conclusion

Many textbook treatments of the twins' paradox assume that the travelling twin remains in an inertial frame for almost the entire journey. While this simplification makes the mathematics easier to present, it can also obscure the role of acceleration, which is necessary for the traveller to reverse direction and return to Earth. The resulting asymmetry is often explained using a discontinuous "jump" in simultaneity. In this interpretation, however, the jump is a feature of the piecewise construction of inertial reference frames rather than a physical effect.

As discussed above, the same mathematical framework can be interpreted without introducing this "jump". In the limiting case of purely inertial motion, where no turnaround occurs, both twins remain in inertial frames throughout the journey, and the comparison reduces to a single inertial description. This highlights the importance of using a physically consistent coordinate construction rather than relying solely on idealised spacetime diagrams. Figure 3 illustrates that the traveller's proper time is shorter than the coordinate time ($\Delta t_{0}$) from the beginning of the journey. From this viewpoint, the difference is not created by the turnaround or by changing reference frames, but is a consequence of the traveller's motion. Even if the traveller never returns to Earth and continues indefinitely at constant velocity, the proper time accumulated along the traveller's world-line remains shorter than $\Delta t_{0}$. Consequently, the coordinate time of the Earth's frame appears dilated when viewed from the traveller's frame. In this interpretation, the term "time dilation" refers to the apparent expansion of the rest frame's coordinate time, while the traveller's elapsed duration (proper time) remains shorter.

Both twins divide the journey into the same coordinate intervals: five units of time and four units of distance for each leg. What differs is the size of the units used in each reference frame. Although the observers use different systems of measurement, the underlying spacetime events remain the same, just as 1 inch and 2.54 centimetres describe the same physical length using different units. Their measurements can therefore be compared consistently, provided they are first expressed in a common reference frame through the appropriate Lorentz transformation.

The travelling twin only appears younger because less proper time has elapsed along the traveller's world-line during the journey. Once the trip is complete and both twins are again at rest relative to one another, physical quantities such as distance and coordinate time are again defined within the same inertial frame. The "age difference" therefore reflects the different durations accumulated along the two world-lines rather than any permanent alteration of time itself.

Consider a stick submerged in water: it appears bent due to the refractive medium, but when the water is removed, the stick is revealed to be straight. Similarly, in Special Relativity, acceleration and velocity act as the "medium" that distorts measurements of length and duration. When motion ceases and the "medium" is removed, these measurements return to their inertial values. The traveller twin's clock records a shorter duration (and never recover), but this is a record of the path taken through spacetime, not a fundamental alteration of the time dimension itself.

From this perspective, much of the apparent paradox arises when measurements from different reference frames are compared without first applying the appropriate Lorentz transformations. Mixing quantities defined in different frames can produce apparent contradictions that disappear once all measurements are expressed consistently. Likewise, using oversimplified or geometrically inconsistent Minkowski diagrams can make the paradox appear more mysterious than it is.

Ultimately, the apparent paradox arises only from an incomplete comparison of measurements. Once all the relevant data are analysed consistently within the appropriate reference frames, the paradox disappears.