There is no velocity till the reference frame is specified. In the reference frame connected with the object its kinetic energy is naturally zero.  So, you talk about the kinetic energy as perceived in a different frame.

The rest is mathematics. You may do it explicitly or remember that this is the property of the function cosh(x).

Generally, having the dispersion as a  function E(p), the mass is defined as the second derivative at p=0.

You missed at p=0. There is no p=0 in the ultrarelavistic case. In that case, your reference frame moves very fast relative to the object. p^2c^2 >> m^2c^4 and the term with mass is discarded.

that's what relativity is about. Sun circles Earth if your origin is at Earth, Earth circles Sun if your origin in at Sun, Earh makes funny movements in the sky if you are at Mars

Very good responses all.

First I should say there is a preferred reference frame for velocity defined by CMB equal in all directions at 2.7 degrees Kelvin. True, but unpopular, so unless I define some other frame that is the one I intend.

Second the math is simple enough but can be made as complicated as you like. I am asking about the physical make up of vacuum space and how it is expected to respond to an extreme kinetic energy. There are clues in the ratio T/pv, but other clues are not discovered yet. So far the choices appear to be that either an event horizon closes like a black hole, the stress energy elongates to a worm hole, or an infinity occurs in the capacity of space to contain energy and the accelerated vehicle converts into light energy. In research everything reaches a limit. I'm asking for opinions of the limits in space.

Next if you combine the conventional equations you get a fairly simple integration that eliminates velocity explicitly and contains SR.

pc = Ev/c

dE = v dp

(Ev/c)dE = (pc)vdp

dE2 = c2 dp2

GR begins to enter in if light speed is allowed to vary with stress energy, provided Lagrangian density is applied to local space. The full treatment of GR is not employed yet in this way.

About the ultra-relativistic case where we have E = cp, either total energy is so large that mass energy becomes insignificant, or mass energy mc2 converts into part of pc.

An indication is found in energy momentum equation.

E2 = (mc2)2 + (pc)2

The absence of a middle term [ 2 (mc2)(pc) ] on the right side suggest that we have to choose, either mass energy converts into momentum energy as velocity approaches light speed, or the system contains less energy than was put into it.

Mass is never observed moving at light speed. Many possibilities are calculated, and most of them predict bad consequences for the traveler who is not prepared.

In the question at low speed [ T/pv = 1/2 ], what does the other half of pv represent at low speed, and how does that representation diminish as velocity approaches a maximum [ T/pv = 1 ]?  One constraint is missing.

@Remi

Obviously, dE2/dE2 = 1 by definition.

But, the issue is not that. The issue is that the relativy theory ensures nothing happens with the physical body when viewed from a different frame.

@Jerry

There exists a frame in which you are  ultrarelavistic. How does that feel?

Thanks to everyone for comments. Notice that I did not originate any of the math I used. It is just a set of quotations from many standard text books, not even upper level courses. The inference is mine that mass energy converts to momentum energy as speed increases. This is subject to testing but needs a deep space expedition to accomplish it.

Ultra relativistic cases are the ones I am exploring. My question relates to the testing of assumptions and status quo in science. Even from an intermediate level of math, it is possible to raise a question about energy and momentum.

From the equations it can be proved that if light speed is constant, mass is also constant. This is SR.

My other threads are going beyond Einstein but building on his work that stress energy curves space. This is what he asked us to do in his last published work, putting the kinetic energy into the curvature. For me it suggests local light speed increases around a fast moving large object such that the object never exceeds local light speed. This is a warp field induced by kinetic energy.

There is one missing equation, and several possible candidates, but limitations on the possibilities. So far I have found two types of solutions. One type seems impossible physically in that energy reaches a maximum and then decreases as speed continues to increase. The other type is that velocity eventually reaches local light speed in the warp field when effective mass goes to zero and a catastrophe occurs for the traveler who is not expecting it.Things like strength of materials and ability to function are involved. Where the two types meet, there is calculated an infinity of energy and speed which seems unlikely in view of other limitations found in research.

Where T is pv near light speed, but only half of pv at low speed, I believe there is something important hiding in the other half of pv at low speed. It relates to  what type of equipment can be built, and how it can be operated at ultra relativistic speed.

 Remi Cornwall, so many unconventional interpretations are made and not a single deviation is done from the set of conventional equations. This is why I chose the lower level math. Everyone can understand it. There is a more elaborate version in upper level math, but few people understand it, and nothing comes from it but an argument of two people that no one else reads.

One item of stress energy you objected to is well established in QFT from Lagrangian Density in the Klein Gordon Equation. "warp field induced by kinetic energy." Of course GR and QFT are not in agreement, but I have identified the root cause and proposed remedies which are found in my unconventional threads. Albert Einstein in his last published writing identified kinetic energy fields as an unfinished part of his work and asked his readers to continue it to a unified field theory. Non-sequitur does not apply here. It is just slow progress and long overdue.

Reference: Hans de Vries, 2009 Book: Understanding Relativistic Quantum Field Theory, published by Physics-Quest.org 2013, chapter 22 and equation 22.43. The kinetic energy is expressed as a field scalar dot product of two field vectors.

http://www.physics-quest.org/Book_Chapter_Lagrangian.pdf

 I did reject the calculated cases where "energy reaches a maximum and then decreases as speed continues to increase" although they could have been postulated as time going backward, which I didn't do because it is not helpful for the question of this thread.

Warp is widely published now since NASA broke the silence, although it was forbidden for several decades in published science. Where are the apologies to those who suffered indignities for being a little ahead of their time?

The other items you object to are consistent with the equations I used from main stream science. You cannot prove my interpretations are wrong unless you can discover the missing equation beyond the SR constant mass and constant light speed at low energy. In fact I have not chosen just one interpretation, but have displayed several possibilities, pending the discovery of the missing equation.

Consider how important are the known equations.

E2 = (mc2)2 + (pc)2

pc = E(v/c)

dE = v dp

Then consider that the set is not complete and there is another equation undiscovered and it is as important as these in the high energy range.

So the effort Is worth while and all comments are helpful in defining the questions and shaping the answers. 

I'm surprised you didn't object to the system that has less energy than was put into it. That was the first thing that caused me to investigate this apparent backwater of science. Beyond your many objections I believe you are on the right track and may eventually discover something more important.

Further calculations with squeezed quantum states and scale relativity have concluded that in a rigorous calculation T/pv remains at one half over the velocity range zero to light speed. In the rigorous form it is not described by the function.

T/pv =/=  [ 1- ( 1 – v2/c2 )(1/2) ] / (v2/c2)