The possible convergence of the integral that relates proper time of an accelerated clock with its inertial time motivated a preliminary discussion in a letter to Il Nuovo Cimento in 1985. Two alternative points of view were contemplated in the above mentioned letter to solve the problem. Please see the attachment.

I think it means that the inertial observer will disappear behind a horizon for the accelerating one. For example, it is known that such a horizon exists in the Rindler metric. Now this is not quite the same problem, because any finite proper time of the Rindler observer corresponds to a finite time of the inertial observer. Which shows, however, that the proper acceleration must increase for the integral to converge, because Rindler observers accelerate at constant proper acceleration.

Yet, if the acceleration increases strongly enough with time, then from the point of view of the inertial observer time dilation will be so strong that the accelerated observer never reaches the limiting value of proper time. He will "see" a freeze of the motion of the accelerating observer. (This is, by the way, what happens in Schwarzschild coordinates, for a freely falling observer who according to an inertial observer never crosses the event horizon of the black hole. For the falling observer, the crossing event is at finite proper time; all proper times beyond that point are never accessible to the external observer.)

From the point of view of our accelerating observer (if he survives the ever-increasing proper acceleration), the critical proper time can be exceeded, but the inertial observer has disappeared behind a horizon backward from the direction of acceleration. The interesting thing is that for the inertial observer the "horizon" behind which the accelerating one disappears is at infinity. So for the proper times beyond the limit value, the accelerating observer must indeed be outside the universe of the inertial one. But he left it through the time direction, so to speak. Moving infinitely long at almost the speed of light, he then is at infinite distance, too.

Thank you Professor Kasser for you answer to my question.

As you say, the integral that gives the proper time as a function of the inertial time could converge for suitable speed histories of the accelerated clock on its path. Then, while the inertial time tends to infinity, upper bounds for proper time could appear, such that for the inertial observer the accelerated clock would take infinite time to approach to the “proper age”. Admitting that the proper time is not bounded, part of the evolution of the accelerated clock is beyond the possibilities of description accessible to the inertial observer: his time has run out. Births and deaths, explosions and implosions occurring next to the accelerated clock for proper times greater than the above mentioned upper bounds wouldn’t be facts on which every observer in the universe (inertial or non-inertial) could agree. And yes, this behavior reminds us what people think that happens during the fall of a massive particle towards the event horizon of a black hole.

But the event horizon of a black hole is a property of a structure intrinsic to space-time in the framework of general relativity. Here we have a structure depending on the particular kinetic behavior of the accelerated observer, as you say a kind of Rindler's event horizon for a non-uniformy (proper) accelerated clock.

Please see in the attachment a preliminar research about the behavior of the proper acceleration in order to have a finite proper time for an infinite inertial time.

Dear Roberto,

I don't understand why you complicate this question of the time transformation without solving the integral in reasonable conditions. The solution to your integral for the a proper time would be

τ (t)= (c/a)arcsinh(at/c)

when the initial conditions τ (0)=0 and v(0)=0 and assuming that you have an external constant force F per unit of mass producing a uniform acceleration a, which I understand are the conditions that you were putting your question.

1. No general relativity is necessary at all in this case and only special relativity can give you a convergent solution.

2. Obviously you don't need to go to other universes because the timelike curvas always are well defined without entering singularities.

Dear Daniel:

Thank you for your interest in the question. I am already aware of the case of uniform proper acceleration and events horizons in special relativity (for example in pages 21-22 of the book by Hobson M.P., G. P. Efstathiou, A. N. Lasenby, 2006, General Relativity, New York: Cambridge University Press) that you put in your answer.

You say: " assuming that you have an external constant force F per unit of mass producing a uniform acceleration a, which I understand are the conditions that you were putting your question."

However, these are not the conditions of my question, as can be seen reading the attachments.

The question was posed in the framework of special relativity (although the letter of 1985 to Il Nuovo Cimento that I attached to my first answer to the question was classified as general relativity).

In my opinion, the above answer of Prof. Dr. Kassner highlights the main points of the problem.

Dear Roberto,

What are the conditions that are out of my solution? Do you need general relativity for solving a problem of transformation of the proper time using inertial observers?

Dear Daniel:

Your example of a rindler observer is OK. We don't need general relativity to describe an accelerated clock relative to an inertial frame. Moreover, you can use special relativity in general frames, working in flat Minkowsiki space ( as is carefully developed in Eric Gourgoulhon book , Special Relativity in General Frames, Berlin: Springer,2013).

Dear Roberto,

Rindler coordinates only generalize the formula given by the time to spacetime where a constant proper acceleration is introduced. In your question you only spoke on the time transformation in all that I have reading. Thus I don't understand what are you saying.

Obviously, from my humble point of view, there are not any kind of paradox and your equations are easy to be solved in a straightforward form without considering the absurde necessity of having another universe for solving the integral instead of making speculations.

I don't understand the issue of your question. What is the problem please?

Please read Dr. Kassner answer-.

Dear Roberto,

My question is mainly due to read your question and the Kessner's answer that you have accepted so well, although you were asking a very different thing :

I think it means that the inertial observer will disappear behind a horizon for the accelerating one....

But immediatally Kassner follows:Now this is not quite the same problem, because any finite proper time of the Rindler observer corresponds to a finite time of the inertial observer (which is obvious and that is pure mistake because finite time doesn't exist in this relativistic context).

And the fantastic end that perhaps you looked for:

The interesting thing is that for the inertial observer the "horizon" behind which the accelerating one disappears is at infinity. So for the proper times beyond the limit value, the accelerating observer must indeed be outside the universe of the inertial one. But he left it through the time direction, so to speak. Moving infinitely long at almost the speed of light, he then is at infinite distance, too.

Do you think again that your question and difficulties related with the proper time under a constant proper acceleartion need the Rindler observer and no just to solve one integral?

My question is not related with proper time under constant proper acceleration: in this case there is no problem at all. Please read the question and the attachments again.

Dear R. Suárez, amigo uruguayo,

You are progressing because now you could understand that Kassner said that he is not answer your question but something related as the Rindler coordinates. Congratulations, I thought that you couldn't by yourself face the question that you asked.

What would be the consequences of the convergence of the integral that relates the proper time of an accelerated clock with its inertial time?

Answer: The consequences are not related with your wrong assumptions and they only depend of solving a simple integral that you never did. I solved it for you and it seems that instead of discussing my results or accepting them, you said that I needed to read the answer of Kassner. I did it and I wrote in black what considered that he was just speaking on something not related with your question directly. But it seems that you don't understand when somebody is pulling your leg or speaking without entering in your question.

Please, where is a mention of the convergence of the integral by Kassner? Where is even the solution of such integral (although it is assumed in the time component of the Rindler coordinates)? Where is introduced explicitally the clock acceleration and considering it within the integral that you have presented?

Dear amigo ¿español?

The integral that you wrote in one of.your answer is a well known result when proper acceleration is constant. Rindler coordinates are not neededat all, if the description of the accelerated clock is done relative to an inertial system, as is done in the attachments. Please read the formulation of the question again, as well as the attachments.

Besides ¿amigo? please read again Kassner answer.

You wrote in your last answer and told it to me: "You are progressing because now you could understand that Kassner said that he is not answer your question but something related as the Rindler coordinates. Congratulations, I thought that you couldn't by yourself face the question that you asked. "

Now, afeter reading your comments, it seems to me that you are not progressing in the comprehension of what has been said.

Dear Prof. Suarez-Antola,

It seems that you would need a time frame in such case in Special Relativity. If you don't mind I can present my papers Mallick (1993, 2012, 2014), Mallick, Hamburger & Mallick (2016), which except the first one are on www.researchgate.net/profile/Soumitra_Mallick, which analyses somewhat similar problem but in String Theoretic Econophysics spacextime. I had presented one of the papers, the second one at the World Finance Conference in Brazil in 2012 and had chaired my session on Market Microstructure.

Soumitra K. Mallick

for Soumitra K. Mallick, Nick Hamburger, Sandipan Mallick for NHMHM School.

It is a specific characteristic of the Walrasian Cartesian system in such Econophysical problems with Lorenz trasformation of the cybernetic spacextime, if necessary to accomodate non-Markovian stochastic programmes (ours is pretty general dependant on stochastic processes in general with probabilities) that the actions have closed graph because they are based on equilibrium convergence and limits. Hence, all curvature questions are Newtonian Systems Classification and Systems Integration of vector fields with topological vector diffeotopy of embeddings. Observations are localised hence there are always world sheets for String Matching Fields with possibilities of spectrum resolution but optimal control of observations and actions. String perturbations can be mathematically as well as statistically analysed because of operation of Gauss-Markov Theorem. Thought I would add these explanations.

Soumitra K. Mallick

for Soumitra K. Mallick, Nick Hamburger & Sandipan Mallick for RHMHM School.

A very nice question, and Kassner's answer is beautiful. I would merely like to add a reason, why this need not worry us too much: the distance to the horizon of a particle having acceleration a is of the order c^2/a.

But in your example, a must go to infinity as t does. So if the object thus accelerated has any sized l, the horizon would eventually separate the forward and aft part of the particle, so that major disruptions would arise. The paradox can only arise for systems which are strictly pointlike. This may therefore suggest that the paradox is less bad than might seem at first.

Thank you all for the additional answers.

Dr. Leyvraz, the accelerated clock is strictly pointlike by assumption (it is an ideal clock) as you say.

It seems that a necessary condition for the convergence of the proper time integral when inertial time tends to infinity is that proper acceleration tends to infinity when proper time approaches a finite value from below. The same behavior for proper force. (The attachment "The behavior of the proper acceleration when the integral of proper time as a function of inertial time converges").

So, if this case is excluded, it seems that we have no paradox: the integral of proper time would tend to infinity when inertial time tends to infinity.

Dear Leyvraz,

The question is:

What would be the consequences of the convergence of the integral that relates the proper time of an accelerated clock with its inertial time?

Please, could you develop here where is the possible paradox and what is so beautiful in the Kassner's answer? Thanks.