The Relativistic Photon Rocket
We know that the classical Tsiolkovsky ( Newton ) rocket equation is given by
Hence, the Tsiolkovsky rocket equation becomes
Interstellar Rocket Travel
However, beyond earth's solar system and into interstellar Milky Way space, notwithstanding the means by which any such rocket achieves this immediate goal, the relativistic photon rocket equation necessarily becomes
The Parameters Set Forth for a Relativistic Photon Rocket
The only boundary conditions set forth here are
However,
And therefore,
What this ideal relativistic photon rocket equation says is that it is mathematically possible to directly relate earth observations for time, distance and velocity of the relativistic photon rocket with those of the astronauts making proper time, velocity, etc. calculations if and only if immediate communications are available between earth and the rocket but, of course, never will this be since all electromagnetic propagation of information is transmitted at the finite speed of light, c.
Nevertheless, the following equations express the basic relativistic equivalencies between measurements by earth observers and the photon rocket's astronauts:
§ Proof of ( 1 ):
§ Proof of ( 2 ):
§ Proof of ( 3 ):
§ Proof of ( 4 ): already done. see above.
§ How Relativistic Astronauts See Things ( time, distance, velocity, and acceleration ) Differently From Earth Observers^{∗∗}:
Assume that our relativistic photon rocket achieves from the point of view of the astronauts; then, we now calculate what the equivalent measurements for time, distance and acceleration to some ultimate destination to a star or galaxy will appear to earth observers as follows:
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^{∗∗} note: these examples are used in the future upcoming Relativity Physics and Science Calculator Mac application
However, now try this as we create our relativistic photon rocket with astronauts inside:
What this shows is that at , as observed from earth, neither a photon rocket nor any subatomic particle can accelerate beyond the speed of light and that the mass of the photon rocket as well the total energy of the relativistic photon rocket !!
Equally since , the astronauts will calculate their speed as probably asymptotically stopped in any forward direction towards their ultimate star or galaxy destination!
Lecture by Alex Filippenko, Ph.D.: Special Relativity  Interstellar Travel
§ Now we invoke one of the boundary conditions of the entire photon rocket consisting almost solely of photon energy:
The mass of the astronauts and their payload of instruments such as accelerometer, chronometer, measuring ruler, and so forth, will be judged to be negligible compared to the total mass of our ideal creation for a relativistic photon rocket reaching deep into interstellar Milky Way space.
Therefore,
And for this measurement our earth observers of the photon rocket will calculate rocket velocity as
Finally, the total energy requirement for our relativistic photon rocket to some ultimate star or galaxy destination becomes:
Earth Observers vs. Astronauts v/c = 0.60 and v_{e}/c = 0.693 


Destination ( Galaxy or Star ) 
dl^{ ∗∗∗∗} ( earth distance ) 
dl_{e}^{[1]} ( astronaut distance ) 
dt ^{[2]} ( earth time ) 
dt_{e}^{[3]} ( astronaut time ) 

Milky Way Center  2.5 x 10^{4} ly ^{[a]}  2.311 x 10^{4} ly  4.166 x 10^{4} years  3.333 x 10^{4} years  
Proxima Centauri  4.223 ly  3.903 ly  7.0 years  5.6 years  
Sirius star  8.6059 ly  7.954 ly  14.34 years  11.47 years  
Procyon star  11.4043 ly  10.540 ly  19.0 years  15.2 years  
Andromeda ( M31 )  2.544 M ly ^{[b]}  2.351 M ly  4.24 x 10^{6} years  3.39 x 10^{6} years  

Bondi kCalculus and Minkowski's Spacetime Geometry
( See: Bondi's interpretation of Minkowski's spacetime geometry )
Let's back up a bit and examine what has already been obtained in our examination of the relativistic photon rocket, namely that
Telepresence as an Interim Solution
We here on earth could in the interim begin employing some of the latest technologies for "telepresence" whereby micro robots ( micro  bots ) could in every practical sense, be flung into deep intergalactic space, way beyond Milky Way and approaching say one  tenth the speed of light, to send back "virtual presence" of sight, sound and, yes, even touch to earth observers of far, far distant suns and planets. Why not? But this aspect of human exploration should begin sooner rather than later in order to reap this sort of virtual experience by the middle of the next 22nd century.
Summary
Thus, the relativistic astronauts will always consider themselves closer in distance and time to their ultimate star or galaxy destination than will the earth observers determine time and distance for this same relativistic photon rocket. This phenomenon is due also no less than to a doppler effect of compressed crest  to  crest wave frequency for incoming [ electromagnetic ] light as our relativistic astronauts take measurements of the incoming electromagnetic light  i.e., measuring shorter time intervals for higher electromagnetic wave frequency as compared to earth observers. In other words, the relativistic astronauts will in effect be using the compressed crest  to  crest wave frequency of the incoming [ electromagnetic ] light as their compressed ruler versus the earth observers for taking time and distance measurements against a "fixed" starry background!!
Philosophic Considerations of the Human Condition
The Relativistic Photon Rocket demonstrates how time and distance judgments are different for photon rocket astronauts and earth observers for the same ultimate star or galaxy destination. Even more importantly owing to the "human dimension" of all this relativity mathematical physics, is the fact that we humans live at the most 80  90 earth  years [ most of which is taken up with socialization, education, etc. ] and yet the distances even within our own Milky Way space are so tremendously vast, that astronauts travelling at even 50% the speed of light would not live long enough to do anything meaningful on some other planet circling whatever distant star to which they were aiming. In other words, there's the 4th dimension of relativity, probably even a 10th or 11th quantum dimension, but the entire matter of the "human dimension" is what ultimately will limit direct human exploration of interstellar and intergalactic regions of the cosmos. And this is neglecting for the time being any biologic revolution necessary for such galactic travel thru radiation fields, dark energy fields, and so forth. But should we humans be enthusiastic for interstellar exploration of Milky Way space? Definitely yes!! But it must first be accomplished by means of instruments and robotics, not human folks. And, finally, these relativistic photon rocket equations should make all humans further aware how deeply important is planet earth for human survival as even the nearest of the stars are presently beyond human reach.
.....................................
Insofar as the human condition is existentially subject to the laws and equations of the external world, we humans will never know instantaneously what is occurring in the farthest portions of the cosmos simultaneous to events coterminus to our immediate experience. This fact of human existence is owing precisely to the unquestionable fact that the speed of light as well as other electromagnetic propagations of information is finite and hence not instantaneous. Although Plato never imagined this conundrum when he postulated his cave for human knowledge, it is indoubtedly also true that we humans live in a sort of "experiential bubble" and that our greatest means to expand our knowledge of the external world to whose laws and equations we are subject, lies in mathematical physics and also in the mathematical equations of chemistry. All other means of knowing and understanding the external world are bogus and only express inner human needs of emotion. See: "Language, Truth, and Logic", by A.J. ( Alfred Jules ) Ayer, first published V. Gollancz, Itd,, London, 1936; still now available from Dover Publications, Inc. ISBN: 0486200108
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Finally, as far as the human condition in the cosmos is concerned, it is truly existentially tragic. Even when we humans make contact with other intelligent beings in the cosmos, all is doomed no matter what. Why? Because of the expansion of the cosmos beyond and into the Ultimate Void on the other side of the cosmic horizon.
So, as the cosmos expands and becomes ever more and more distant from earth observers, we humans and any other intelligent beings in the cosmos will increasingly see a flattened  out light wave of whatever is "out there"  i.e., in other words, we humans and any other intelligent beings will see less and less until we will finally see nothing. A flattened  out light wave will contain zero energy and hence zero amount of information. The focal length of any telescope will have to equal in physical length the diameter of the entire expanded cosmos in order to see anything which, by definition, will be impossible by any stretch of engineering! Hence, the night sky goes black since all cosmic matter will have achieved the speed of light and thereby becomes pure energy on the other side of the cosmic horizon; no telescope will ever be built to capture the flattened  out light wave; all energy in earth's vicinity is completely and totally dissipated; and, finally, the ideas of velocity, distance and time will disappear from human understanding as there will be no need to measure infinite blackness with infinite nothingness to observe!!
This then is the tragedy not only of human existence in this cosmos but also the tragedy of other intelligent life in the cosmos. In the physical external world, that is.
But long, long before this eventual cosmic end, our sun will have burnt up and with it earth itself will burn up from the supernova that the sun will have become in some 4 billion earth  years, plus or minus a 100 million earth  years here or there.
Carl Sagan and the Golden Record flying Endlessly into Interstellar Space
§ References:
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