There isn't any relation between time and mass in speical relativity: mass is a Lorentz invariant, the same in all reference frames, time is the fourth component of the position 4-vector. Similarly, energy isn't a Lorentz invariant-it's the fourth component of the energy-momentum 4-vector. So the answer to the question is that the observed mass, which is Lorentz invariant, doesn't increase-or decrease-under Lorentz transformations. The correct formulation is that energy and momentum transform under Lorentz transformations as a 4-vector, so that E^2-(p c)^2 = (mc^2)^2 = (E')^2-(p'c)^2, where (E, p) and (E', p') are the 4-vectors in different frames, that are related through a Lorentz transformation. The expressions for the energy and the momentum, that contain the relative velocity between the two frames, express just that and nothing more: they relate the values of the energy and momentum in one frame to the values in a frame, moving with velocity beta=v/c with respect to the other, which gives rise to the factor 1/sqrt(1-beta^2)=gamma.

From first-principles consider a constantly accelerated box of moving particles or photons.  Then show that the accelerating force on the box (hence the inertial mass of the system) depends on the internal momentum (relativistic mass) of the contents.  The relativistic mass/energy relation E^2 - (pc)^2 = (mc^2)^2 is the basis.  In simple terms, the equation tells us that with no momentum any mass will have a rest energy E=mc^2.    That is the key to the understanding in that it tells us that any mass is exactly equivalent to a quantity of energy.  So there is really no difference between the two.  going back to E^2 - (pc)^2 = (mc^2)^2, if you accelerate a mass, i.e. increase it's energy, the mass goes up by the same amount because they are one and the same.  In the limit this can continue until the the velocity reaches c, the speed of light.

Well, I think the particle mass with respect to its own frame of reference does not increase, since the particle is at rest with respect to itself. 

However, with respect to other frames of reference, the increase in mass just comes out of the derived equation for conservation of energy and momentum. 

E=mc^2/sqrt(1-v^2/c^2)

The denominator in the equation is what causes the increase in mass, since as v approaches c then the denominator approaches 0 which given infinite for the right hand side. Since the energy has to be conserved then the mass increases so as to keep the ratio constant. The momentum is conserved so, there should not be an increase in potential energy.

Of course, there may be more complex cases and theories, but I hope this helps you.

There isn't any relation between time and mass in speical relativity: mass is a Lorentz invariant, the same in all reference frames, time is the fourth component of the position 4-vector. Similarly, energy isn't a Lorentz invariant-it's the fourth component of the energy-momentum 4-vector. So the answer to the question is that the observed mass, which is Lorentz invariant, doesn't increase-or decrease-under Lorentz transformations. The correct formulation is that energy and momentum transform under Lorentz transformations as a 4-vector, so that E^2-(p c)^2 = (mc^2)^2 = (E')^2-(p'c)^2, where (E, p) and (E', p') are the 4-vectors in different frames, that are related through a Lorentz transformation. The expressions for the energy and the momentum, that contain the relative velocity between the two frames, express just that and nothing more: they relate the values of the energy and momentum in one frame to the values in a frame, moving with velocity beta=v/c with respect to the other, which gives rise to the factor 1/sqrt(1-beta^2)=gamma.

Thank you all for the nice answers.

How about the potential energy relative to certain frame is it affected by accelerating the particle to relativistic velocities?

First due to the mass- Energy equivalence, mass is not a Lorentz invariant, rather rest mass assumes as an invariant,in  the well known equation E2 = P2c2 + m2c4, obviously m stands for rest mass,  mass or total mass can be calculated by E/c2 . Therefore mass of a body in especial relativity depends on the relative velocity between observer and rest frame of that body : m= m0( 1- v2/c2)-1/2 where m0 is rest mass. Second, Dear Sadeem, we know that apart the especial relativity , the generator of total energy ( and its mass equivalence) is a Killing vector ∂/∂t that denotes the time translation group generator,in this sense but not strictly ,  i guess you may inspire the connectivity between Energy (and equivalent mass) and time by this presumptive view.    

@ Alexei, Rest mass( also known as intrinsic and proper mass) is the corresponding component of "proper time" in energy momentum four vector, mass (relativistic mass)in general form varies with the speed of particle as the experiment  on high speed electron have been shown. therefore relativistic mass denoted by E/c2 for a free particle. these concepts can be reviewed in all especial relativity related texts and also Goldstein's mechanics (2nd edition pp 308,309).

Just the total Uni-enegry has the property of conserving. Who knows what is mass or time? The classics says the mass is a category, time is stiil independent parameter + Galileo transform + minimal set of axiomes = mechanics. Let the time be a function of space within infinitesimal invariant interval -> we've got a new phisycal vision, as an eigen inclusion into the classics. Then, interpretations follow, but who knows what is mass?

@ Alexei , For a reliable reference (and not historical or high school level ) see the latest reference book in Relativity of W.Rindler : (Relativity SPECIAL, GENERAL, AND

COSMOLOGICAL Oxford press,2006, pp 109,110 ) , this clarifies the above mentioned concepts in Relativity, of course in QFT there is another story because the main context is about interactions and not a free particle.

As far as I can see the only relation that exists directly between time and mass is in relativity, with  M = Mo   dt/ds. ,where t is the observer's time and s is the proper time.

M is called the relative mass and Mo is called the proper mass.  Interpretations and conclusions must be drawn carefully.

Aragam Prasanna, show me definitely what this formula "M = Mo dt/ds" gives pragmatically, but if the time is discrete rate, which would be the time quant without any derivative? 

As Stam said, mass is assumed invariant in the theory of relativity. Some people have used the rest mass and moving mass as an alternate formulation but the results are one and the same.

But in general relativity the invariance of mass is problematic; in fact, it turns out to be impossible to find a general definition for a system's total mass, only the derivative of energy with respect to time is conserved. How can we understand this conflict with special relativity?

I refer you to the following link in Wikipedia as an example

http://en.wikipedia.org/wiki/Mass_in_general_relativity

@Aragam Prasanna and Dmitry Kovriguine.

Both of you are close to the correct answer. Time is strictly a periodic phenomenon. T=1/f is the only fundamental relation that relates imaginary time to physical oscillations (frequency). You can never measure time without the help of an oscillator. The relation M = Mo dt/ds is more fundamentally defined in the works of Prof. Louis de Broglie, which is M = Mo f/fo.

@Sadeem Fadhil.

The observed mass increase with particle speed because with the increaed speed, particle's kinetic energy increase. From planck quantum hypothesis E=hf, we see that f will increase with increase in E. Proper time is related to proper frequency fo. In the rest frame of the particle the rest mass is Mo and increase in kinetic energy is not felt in the rest frame of the particle so fo and Mo does not change. So potential energy does not change. Only kinetic energy and the relativistic mass M change when viewed from another rest frame.

If you have time, please look at this link - http://www.physics-in-5-dimensions.com/5th-dimension-defined.html - which looks at an example based on two space craft travelling initially side by side in space with a set constant velocity c. Each space craft appears to be "at rest" to the other craft. If one space craft is made to change direction, the two space craft separate and have a relative velocity v between them. However the constant velocity c has not changed for either of them in the direction of their respective motion. The relative velocity v observed is therefore only dependent on the angle between their paths with constant velocity c. The example continues (with the "Further development ..." link) to show how Einstein's relativistic mass relationship can be derived from this example. Now you may not agree with my theory or ideas in this example, however I am sure you will gain an interesting insight into the relationship between time and mass, i.e. mass doesn't change as far as each spacecraft is concerned and if the constant velocity c (the 5th dimension in this case) is ignored, then you have to arrive at Einstein's mass relationship in order to conserve energy when working with our standard 4 dimensional universe. 

To Sadeem Fadhil

Further to my previous answer a week ago, I would like to return to your original question and pick up the issue of the “potential energy”.

A relative easy way to explore some of the results of the theory of Physics in 5 dimensions is to look at the resulting expressions of physics alongside their equivalent expressions of Classical Physics. The following link shows such a list of expressions:

http://www.physics-in-5-dimensions.com/keysubjects/AlanDennisClark-KS-5th-dimension-expressions.pdf

The first page identifies the parameters used and the second page lists the different expressions under various headings including potential energy. You can see that for low relative velocities (v4) the expressions of Classical Physics and Physics in 5 dimensions give approximately the same numerical values.

The energy expressions demonstrate the role of potential energy in Physics in 5 dimensions and can be compared to the Classical Physics expressions at the same time. These expressions open the door to new objective views of physics offering other perspectives of the unaltered fundamentals of physics.

A forgotten fact about the whole issue of relativistic mass is that the first experiments that revealed a mass increase with velocity were carried out by Walter Kaufmann at the beginning of the 20th century.

http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN252457811_1903&DMDID=DMDLOG_0025

He carried out his experiments by inducing varying amounts of kinetic energy into electrons.

When the trajectories of the moving electrons were not deflected, (observations made by means of a bubble chamber) he found of course that the total longitudinal inertia of the particle involved the energy corresponding to the rest mass of the electron plus the total amount of the added kinetic energy provided to the particle.

But when the trajectories were deflected with sufficiently high velocities, he discovered that the transverse inertia of the particle involved less energy than the sum of the energy corresponding to the rest mass of the electron plus the total amount of kinetic energy provided to the particle, which gave rise to the debate regarding "longitudinal mass" and "transverse mass", which led to the conclusion that mass was of electromagnetic nature (refered to as "electromagnetic mass" by Poincare).

Close analysis shows that only part of the kinetic energy provided to the particle was involved in the transverse inertia component.

Moreover, further analysis shows more precisely that at any velocity, exactly half of the kinetic energy provided converts to a momentary velocity related "relativistic" mass increment, which means that the translational half of the kinetic energy provided is impervious to transverse interaction while propelling the total amount of energy captive in the rest mass of the particle plus the momentary velocity related relativistic mass increment.

You will find that the following equation derived from these considerations allows calculating the full range of relativistic velocities up to the asymptotic light velocity:

v  = c (sqrt(4ax+x^2))/(2a + x)

where a is the energy in joules contained in the rest mass of the electron (8.18710414E-14 joules) and x is the kinetic energy provided in joules.

c is of course the speed of light in meters.

You will also find that the same full range of velocities can be obtained by using the wavelength of the energies involved:

v = c (sqrt(4ax + a^2)) / (2x = a)

where a is the electron Compton wavelength (2.426310215E-12 m) and x is the wavelength of the total amount of kinetic energy provided to the particle.

Finally, it is well known that the velocity related so-called momentary relativistic mass can be calculated with m(rel) = m(rest) multiplied by the Lorentz factor (otherwise known as the gamma factor). you will find that the following equation gives exactly the same relativistic mass for any velocity, but without any need to use the velocity.

m(rel) = m(rest) + K / (2c^2)

where K is the total amount of kinetic energy provided to the particle, only half of which is required to obtain the relativistic mass, which confirms that half of the kinetic energy provided converts to the velocity related momentary mass increment.

My bad. The second equation is of course:

v = c (sqrt(4ax + a^2)) / (2x + a)

Greetings to you all.

Mass is certainly relativistic, as described by E = mc^2 times Lorentz factor gamma, and as found in experiment with the LHC (numerous papers on this). Time, on the other hand, does not 'dilate' except as an artefact of our perception due to increasingly Lorentz-adjusted frames of reference. The problem lies in humans being unable to set a relativistic clock, that adjusts with the energy scale.

Time actually shortens with energy, such that mass and time are proportionally inverse on 'running' with the energy scale. [My latest paper covers this in some detail - not yet published.] Lorentz transformations cancel for mass x time. This means there is a mass-time relation which is relativistically invariant, being mt = h/c^2.

This has profound consequences for action at the so-called event-horizon for a black-hole. The speed of light is still c, but as mass approaches infinity it is seen that time approaches zero (an almost infinite amount of 'our' time occurs every second of BH time).

There are numerous other relationships where Lorentz transformations cancel, showing that a fully 'running' model dispenses with them, and is 'relativistic' at all times and energies.

Regards,

Hello, Terry,

Physics starts with invariants. In the beginning, there was an interval, and the interval was the same in all inertial frames of references. The fact that a signal has limited by velocity $c$ had suggested a form of kinematics. It turned out that the signature of the desired frames of references is depressing, namely; three-plus and minus. Unusual, or rather, unrepresentable for ordinary perception, situation. Let the mass be an attribute of a material point, as in classical mechanics. Let kinematics reserve the measurement of intervals, distances and times. Kinematics has no relation to mass in any physics. Now we need to build dynamics. There is no way to do without energy and momentum. It is necessary to find conservation laws in dynamics. It is not possible to build dynamics in direct analogy with classical mechanics - the signature is no pretty nice. People found a way out, invented a 4-vector, calculated the scalar product, and, found a sought conservation law. Is it worth discussing the attribute of a material point, namely mass? Why connect it with kinematics? The gamma-factor is a purely kinematic construction, not connected in any way with the mass from the moment of its birth.

Hello Dmitry,

Firstly, I should have mentioned (above) that for two observers measuring the same event in different reference frames, of course relativity will still exist and is relevant. The problem with 'clocks' is that we assume they should tick at a rate compatible with our determination of time. Relativity will occur at different 'clock' rates dependent upon the energy density at the event.

If you are looking at kinematics, then this is really a gravitational approach, where gravity is the energy-dependent geometry of space-time, through which events occur. Everything (in my opinion) is connected at some level, so kinematics actually does have a connection to mass (actually the mass-wavelength duality).

Energy and momentum connect seamlessly with mass, gravity, and space-time (better expressed as the Higgs Condensate).

I'll leave you with this teaser: surprisingly, although the speed-of-light 'c' is constant, and will always be measured as 'c' within any frame of reference by an observer in that frame, how is it that 'c' is different for each wave-particle duality, even though all wave-particle dualities are themselves mathematically equal?

(This is an active area of research.)

Dear Terry,

At school, I've learned there are two theories of relativity. The first theory is simple for young people while the second theory with metric tensors is hard to study in a secondary school. Both theories start with kinematics. This means to listen to a story first on geodesic curves that need no energy and masses, no forces. In the first case, the story reduces to a linear transform generalizing Galileo. In the case the second, we should deal with the differential geometry in noninertial frames of references.

Then, the Dynamics appears. This is completely Newtonian mechanics though with known restrictions because of those geometry and chronometry, or simply, kinematics.

Afterward, there appear some formulae of type E equals to m with c squared, and people fall in different phantasies of what depends on what. I would like to ask you if the dynamics influence the kinematics? If yes, let me see this way.

I also believe like you that the World is rather interesting than these theories predict. But I cannot agree with interpretations of partial results gluing through the energy-mass-momentum with kinematics while separated, namely; kinematics and dynamics are separated. Is not it? If no, show me that way.