I propose a non-relativistic explanation and calculation of the time dilation in terms of the difference in observational times for an object that is alternatively inward- and outward-bound relative to the observer, based on the following thought experiment:

Imagine a spaceship coming from planet X to Earth with velocity V. The time during which the spaceship can be observed from Earth equals t t_{c}, where t is the travel time of the spaceship (the time it takes for the spaceship to come to Earth from planet X), and t_{c}is the time it takes light to come to Earth from planet X; this expression of the observational time is explained by the fact that the spaceship is following behind its own light.

Alternatively, imagine the same spaceship going from Earth to planet X with the same velocity. In this case, the time during which the spaceship can be observed from Earth equals t + t_{c}, as it takes t for the spaceship to go to planet X, plus it takes t_{c}for the light to convey the image of its landing at that planet to observer on Earth.

So, we have two different observational times for an alternatively inward- and outward-bound object travelling the same distance with the same velocity. In order to reconcile the difference between these two observational times, we find out the mean observational time (t_{m}) as the geometric mean of them:

t_{m}= √ (t t_{c})(t + t_{c}) = √ t^{2} t_{c}^{2}

To find out the factor for time dilation, we divide the mean observational time by the travel time:

t_{m}/t = √ 1 t_{c}^{2}/t^{2}= √ 1 V^{2}/C^{2}

Therefore, in result we have the same factor for the time dilation as the factor used in the special theory of relativity, i.e. the reciprocal of the Lorentz factor (√ 1 V^{2}/C^{2}).

More generally, this method of calculation of time dilation can be applied to the objects moving in circles and all other non-rectilinear trajectories in the following way:

As this method is non-relativistic, an observer can be placed at any point in space, so for the purposes of calculation of the time dilation of a moving object, we substitute the trajectory of its movement, no matter how complex it is, for a straight line (in a physical sense, its possible because time dilation does not depend on the trajectory) and place the observer in the middle of the distance L = Vt, where V is the velocity of the object and t is the time during which the observer would observe the object going distance L if he saw it immediately, without delay caused by the finite speed of light.

The effect of time dilation results from the difference of times of observation of an object that is alternatively inward- and outward-bound relative to the observer placed on the straightened trajectory of the object's movement:

ε = t_{m}/t_{a}= √ (t_{a} t_{c}/2)(t_{a}+ t_{c}/2)/t_{a}^{2}

In this formula:

ε is the time dilation factor

t_{m}is the geometric mean of times of observation of an alternatively inward- and outward-bound object: t_{m}= √ (t_{a} t_{c}/2)(t_{a}+ t_{c}/2)

t_{a}is the time during which the centrally located observer would observe an object that is alternatively inward- and outward-bound on the straightened trajectory of movement (going distance L/2), if he saw it immediately, without delay caused by the finite speed of light; t_{a}= t/2 = L/2V

t_{c}is the time that takes light to cover distance L; t_{c}= L/C

Again, as it was in the initial thought experiment, the reinvented formula of time dilation factor is equivalent to the formula used in the special theory of relativity, i.e. the reciprocal of the Lorentz factor (√ 1 V^{2}/C^{2}).

Moreover, this formula is applicable to the gravitational time dilation as well. In this case, we place an observer at the center of gravity and substitute t_{a}for the gravitational escape time t_{e }equaling L/2V_{e }, where V_{e}is the gravitational escape velocity (the reason for t_{e }= L/2V_{e }is that in order to barely escape the gravity through its center, an inertially moving object has to approach that center with the escape velocity).

According to the well-known formula, V_{e }= √ 2GM/r, so the resulting formula for the gravitational time dilation, which is ε = √ (t_{e} t_{c}/2)(t_{e}+ t_{c}/2)/t_{e}^{2}, is equivalent to the formula used in the general theory of relativity (ε = √ 1 2GM/rC^{2 }= √ 1 V_{e}^{2}/C^{2}).

In conclusion, this explanation of the time dilation, substantiated by the respective calculations, suggests that the time dilation is an observational effect caused by the finite speed of light, rather than a consequence of velocity or gravity as such. In contrast to the theory of relativity, the most important feature of light here is not that its speed is a constant, but that its speed is not infinite.