3.1 - Mass, Energy and the Universe's Speed Limit
In many texts on Einstein's special relativity theory mass has been defined in a circular manner. Some such texts have asserted a four-vector momentum in the form of as a premise which doesn't work for massless particles and then defined mass as the contraction of that vector or visa-versa. In order to avoid circularity and to include massless particles and also in order to facilitate a smoother transition to relativistic quantum mechanics this text will take a newer though not unique approach. For example, here there will be two different momentum four-vectors distinguished by capitalization , and . The lower case will be the momentum four-vector of the first kind which is what momentum 4-vector in relativity refers to otherwise unqualified, and the upper case will be the momentum four-vector of the second kind also called the cannonical momentum. This is done in part because a particle's mass will be defined as the contraction of the momentum four-vector of the first kind which is the momentum four-vector referred to in classical relativistic (non-quantum relativistic) texts. The momentum four vector of the second kind is here defined mainly because its elements are what will correspond to quantum operators in relativistic quantum mechanics. (Some authors choose the capitals the other way around)
From experiments or due to quantum mechanics we know that the magnitude of the three component momentum of a particle can be related to a wavelength (whether or not the particle has mass).
and the relation between the three element momentum and the three element k (whether or not the particle has mass) is
Also, a fourth element corresponding to a time coordinate can be related to a frequency (whether or not the particle has mass) and that element times c we will term relativistic energy ER.
ER = p0c
where can be related to the wavelength by
and from we get:
The integration constant will turn out to be proportional to the square of the mass.
Mass, Energy and the Universe's Speed Limit 22
In this text we will start with the premise that this four component definition of momentum constitutes a four-vector that will be called the momentum four-vector of the first kind .
Next consider the introduction of a four-vector potential to which the test particle responds with a charge q. It does not matter at this point if this charge is electric, only that the vector potential to which it responds is a four-vector. We will define the momentum four-vector of the second kind by
and we will call its time element P0 total energy E
E = cP0
As an artifact from physics texts that do not make this distinction, when the potential is not zero one might think of the total energy in special relativity theory E as cp0 even though it is really cP0 which includes the electrical potential energy. At the same time, one would think of the relativistic momentum as pi. As a result one may think of energy E due to containing a potential as something that can have an arbitrary constant added to it, but would think of momentum as something that can not. This superficially seems to draw a distinction between time and space, as energy corresponds to a time element and momentum corresponds to spatial elements. However here where the distinction between momentum four-vectors is made, one finds that there is no such distinction between time and space. This is because it is in that such an arbitrary constant can be added to the potential, and it is also in that four-vector that such arbitrary constants can be added to the spatial components of the vector potential. One can do this so long as one demands that they transform as the coordinates do, as a four-vector.
The special relativistic definition for the mass of a particle given those relations is
This could just as well be expressed as
In the second we define E0 as the relativistic energy evaluated at zero velocity, E0 = ER|v=0. Magnitude bars are included above merely so that the choice of sign convention for the metric's signature is arbitrary.
Though all 3.1.4 are equivalent given the above relations, the definition in terms of the momentum four-vector of the second kind equation 3.1.4a is preferable in quantum mechanics discussions because that is what yields relativistic quantum mechanics. For example, when one replaces the elements of the momentum four-vector of the second kind with the energy and momentum operators of quantum mechanics, and operates that on the wave function it yields the Klein-Gordon equation with the inclusion of a nonzero vector potential
which can also be written
where cPop0 = H
The above definition of mass 3.1.4b, that a mass is rest energy m = E0/c2 , E0 = ER|v=0. or that its is center of momentum frame relativistic energy m = Ecm/c2, Ecm = p0cmc|vcm=0, in the case of a system of particles, is the definition that we will use throughout the rest of the special relativity site where ever the letter m or the word mass is used unqualified. This is the m that goes into the relativistic version of Newton's second law in the form
(four-force = mass times four-acceleration see - Eqn 3.2.3)
This mass is an invariant. It does not change with speed! Equations 3.1.4 is called the mass-shell condition, because they are of isomorphic form to the equation of a spherical shell. Under the above definition of mass, a photon does not have mass. Due to quantum mechanical issues, virtual particles do not tend to have the expected value of energy for a given momentum. So sometimes it is said a particle lays off shell.
The coordinate velocity of a particle is simply given by
We write Four-Vector Velocity or Proper Velocity
where is called proper time, which is just time according to the particle at its location.
Through time dilation we can relate the two
Consider the following expression
3.1 Mass, Energy and the Universe's Speed Limit 23
Next we refer to the definition of mass 3.1.4c to arrive at
Final examination of this reveals the relation between four-vector velocity and the four-vector momentum of the first kind.
We can then from equation 3.1.8 discover the relation between four-momentum and coordinate velocity for massive particles
or write the relation for particles that may or may not have mass
The term is physically associated to the velocity term through time dilation. In the past a few physicists starting with Planck, Lewis Tolman, not Einstein, have miss-associated the term with the mass defining a new kind of mass
This M is then inappropriately called "relativistic mass". In the absence of a potential, the zeroth element of the momentum four-vector is defined as the energy divided by c, resulting in
p0 = Mu0
E/c = Mc
Though much more complicated in the long run, the math is consistent and leads to consistent predictions concerning observation and so one might argue that the physics is therefor correct. But, in keeping with Occam's razor this definition and method must be done away. The m in this method is then inappropriately qualified and called the "rest mass". It is wrong to do this for the following reason. Calling m the "rest mass" infers to the listener that m is not the mass according
24 Chapter 3 Special Relativity Dynamic Implications
to others for which it is not at rest. We have already noted that m is an invariant as it is the same value as calculated according to any frame. It is not just the value for rest. The relativistic mass method also leads to many erroneous conclusions. By that method light has zero "rest mass". For one of many examples, it has been argued that since light is not ever at rest, that the question of whether it has mass at rest or "rest mass" is unanswerable. No. m = 0 is observed as the contraction of a photon's four-momentum according to any frame, not just at rest.
In short the terms "relativistic mass" and "rest mass" need to be done away and the real mass m which is actually observed is an invariant. It does not change with speed. Also, by this, the physically correct definition a photon, or anything that travels at the Lorentz invariant speed c, has zero mass.
p0 = ER/c.
We have also demonstrated the relation between Four-Momentum and Four-Velocity Eqn 3.1.9 resulting in
p0 = mU0.
Putting these together we have
ER/c = mU0
This is the mass - relativistic energy relationship for a massive particle. Now this energy does not go to zero as v goes to zero so we see that a massive particle still has energy even when it is at rest. This tells us that mass is equivalent to rest energy meaning relativistic energy at zero velocity
3.1 Mass, Energy and the Universe's Speed Limit 25
The kinetic energy of a particle is the amount of energy that is associated with its motion only. Therefor
EK = ER - E0
This results in
In relativity theory the stress energy tensor is a tensor that contains information about the density of energy, momentum, stresses, etc.. contained in the space. The energy tensor mass alone is Eqn. 5.1.4
The T00 component of this is
is the mass density when moving with that bit of mass, but because of special relativistic Lorenz length contraction on the local mass the coordinate frame mass density is then
So this becomes
But this is just the coordinate frame energy density.
The simplest consistent general relativistic definition of coordinate frame energy density is then just
For more general stress-energy tensors it is common to define as
If is to be positive then must be greater than 0. For this to be nagative is called a violation of the weak energy condition. More generally speaking a the weak energy condition is
for any timelike vector . Matter may only violate this condition within limits set by the Pfenning inequality.
Other elements have other interpretations. For instance Tii is a flow of momentum per area in the xi direction or the pressure on a plane whose normal is in the xi direction. Tij is the xi component of momentum per area in the xj direction or describes a shearing from stresses. T0i is the volume density of the ith component of momentum flow.
Next we will discuss the concept of system mass. We have seen that for a single particle mass is equivalent to rest energy. Eqn 3.1.14
E0 = mc 2.
For a system of particles the best concept for system mass m is defined as center of momentum frame energy Ecm.
Ecm = mc2.
The system mass does not turn out to be equal to the total or sum of masses mtot of the constituent parts. Instead it is the total energy summed for all of the constituent parts according to the center of momentum frame.
Consider for a moment a Lorenz invariant for the system consistent with the mass shell condition.
psys' = [Ecm/c , 0, 0, 0]
as a four-element vector for the inertial center of momentum frame. Then define Psys as the Lorentz transform of this for any observer of interest.
Due to relative simultaneity psys as defined here is not always equal to the "simultaneous" sum of the four-momentum of the constituent parts when there are external forces acting at various locations on the system. The system mass is defined as the following invariant.
or for the scenario described above,
Considering the time element of equation 3.1.21 restores the relation
Proof that this definition of system four momentum is the same as the sum of the four-momenta of the systems components goes as follows. Start with the sum of four-momentum for an arbitrary frame.
Interchange sum and transformation symbols
The Lorentz transform of each four-momentum is the four momentum according to the new coordinates
But the right side is the net four momentum according to the new system
This proves that for special relativity the system net four-momentum is indeed a four-vector itself and yields
where m is the system's mass, as well as
The reason that the mass of 3.1.21/3.1.22a is not the same as the "total" of constituent masses, mtot , is that the sum of masses of the constituent parts does not always equal the center of momentum frame energy. For example, a system of massless particles have a zero mass shell condition when they all move the same direction while the system has a nonzero mass shell condition when they move in different directions. One advantage the definition of center of momentum energy for mass has over "total mass" is that by this definition, not only is mass an invariant, but this mass of a system is also conserved. Note that the concept of captive mass is equivalent to center of momentum energy m and not the total of masses mtot. In order to increase the system mass m one must increase the total center of momentum frame energy Ecm equivalently. This demonstrates that mass defined by m is conserved in the same way that energy is. Transferring energy from some external matter to change the center of momentum energy of an object will increase its individual system mass, but when you extend the system to include the matter from which the energy was transferred, it will always be found that center of momentum frame energy or system mass m is ultimately conserved.
A sum of invariants is also an invariant and so one could just as well write the total of masses mtot as a sum of the constituent parts. For a system of n particles this could be written
mtot = m1 + m2 + ... + mn
The subscript indicates the particle number. Again, the major problem with thinking of a system mass as this is that this total of masses is not conserved. However, part of the reason this is brought up is that people do tend to think that mass is that kind of sum. This leads to another misunderstanding as to what is meant by "mass to kinetic energy conversion". Consider for example a massive particle that decays into two massless photons. Because system energy is conserved and the system energy for the center of momentum frame is the system mass, the system mass did not change in the decay. What did change was that the energy initially was associated with rest, the rest energy of the particle, but finally was associated with motion, the kinetic energies of the photons. In that light one should really not say there is mass-energy conversion. The energy and system mass for the system is conserved. One should instead say that energy associated with the resting particles of the initial state becomes associated with the motion of the particles in the final state. Since this is cumbersome the term mass-energy conversion is used, but be wary that what it refers to is that a change in the sum of masses can be the cause of the change in kinetic energies of the remaining masses. Just remember that the system mass of a closed system doesn't change or "convert" into anything.
Sometimes it is more useful to define an invariant mass density instead. Just as there are two ways to describe the masses of a system above, m and mtot, there are two important ways of describing an invariant mass density. The first is the in the following relation equation 5.1.4
This definition most closely corresponds to the mass m description of system mass above. It relates the stress energy tensor for matter composed of non-interacting constituents to the four-velocity of the unpressurized "fluid" at any given location.
The total energy for the system is conserved and could instead be defined by the following volume integral
For the example stress energy tensor 5.1.4 this would become
One can also instead define the system momentum from the next integral
For the example stress energy tensor 5.1.4 this would become
One then still can define the system's mass as the center of momentum energy. It is the energy for the frame according to which
psys = 0. So we still have
Ecm = mc2.
The other invariant mass density concept corresponding to the total of masses mtot would be
This kind of mass density is an invariant, but its volume integral is not conserved. This kind of mass is what is meant when one refers to a field such as the electromagnetic field as a massless field, or when one refers to any system as massless. This is zero for any system of massless particles.
Of the two descriptions of system mass, the mass m concept is far more useful.
Refer to Eqn 3.1.11
where was given by
Notice that the energy becomes divergent at v = c for nonzero mass. Thus no matter how hard or how long you push on a mass, you can never impart enough energy to it so that it reaches the speed c. The only way such a thing can travel at the speed c and still have finite energy is if it had zero mass. In that case, instead of a mass energy relation, there is a energy momentum relation from Eqn 3.1.5 resulting in,
E = ER = pc,
where E and p are related to frequency and wavelength.
One might consider the case of a particle that instead of being pushed beyond the speed c, moves faster than c upon its creation. Such a hypothetical particle is called a tachyon. Notice that if v is greater than c then is imaginary. Since imaginary energy makes no physical sense, we would expect that the mass would also have to be imaginary so that the energy(and momentum) would be real.
The primary problem with the existence of such a particle is that it could be use to violate the principle of causality. The principle of causality is simply the statement that effect never precedes cause. Imagine setting up a tachyon emitter and a tachyon receiver at different points along an S frame x axis. Lets say that the signal travels arbitrarily fast so that the event of transmission and the event of reception are virtually simultaneous. Next recall that events simultaneous in one frame are not all simultaneous in others. We could then easily pick a coordinate system to look at the situation in which the event of reception precedes the event the event of transmission. This is a violation of causality.
Worse yet, it then leads to grandfather paradox's. The grandfather paradox is the idea that a time traveler goes back in time and kills his grandfather before his father was conceived. To set up a grandfather paradox with tachyons in special relativity we simply set the receiver in motion away from the
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transmitter and give it a relay transmitter. We call the frame in which it is stationary the S' frame. We also connect a receiver to the S coordinate transmitter. We then program the S transmitter so that in say 1hr it will send a signal unless it's receiver receives a signal. To begin lets say that it receives no signal and so it sends one. The signal arrives at the relay, which is moving away and sets off the relay transmitter. Now the relay transmitter sends a return signal, but the return signal travels back to the S receiver/transmitter setup virtually instantaneously according to the relay's S' frame. Due to the Lack of simultaneity the return signal will be received back at the S transmitter at a time prior to the original transmission. But because we programmed it not to send a signal if it receives one it will now not send a signal. But then there is no signal to receive and so it sends one......
It is sometimes said that Einstein's special relativity theory says that nothing can travel faster than the speed of light. As discussed, what it really implies is that long as we restrict our physics to special relativity, and we wish to preserve causality, information can not travel faster than c. Likewise, as long as we restrict our physics to Einstein's special relativity, nothing with mass can travel at the speed c.
There have been experiments done in which the physicists involved say that they have indeed been able to get electromagnetic waves to propagate information faster than c through a disspersive medium.
In particular the controversy is over gain assisted faster than c group velocity transmission demonstrated in anomalous disspersive media.
If their claims that it was the "information" that has indeed been transferred at faster than c speeds are correct, then SR implies that we can find a frame according to which the reception of a signal at one end of the apparatus precedes the transmission of the signal at the other end. This would indeed be a causality violation and brings to question the validity of the principle of causality and causes us to reevaluate the (im)possibility of a physical grandfather paradox. However, it is conceivable that the universe may be structured in such a way that such a causality violation is attainable, but that grandfather paradoxes will still not be allowed. For example, if one of their disspersive media faster than c experiments were devised in attempt to simulate the relay-transmitter paradox discussed above, one could hypothetically set such a receiver in motion such a medium, but that medium is what determines the speed of the electromagnetic waves according to its rest frame. A relay transmitter sending the signal through the same medium would not end with the signal arriving at a time prior to transmission.
The following is an explanation why the experiments are not completely convincing of faster than c "information" transfer.
As an example, in a 6.0cm medium a laser pulse has been transmitted that transversed the distance at a speed of 310c. This is a group wave speed, not a phase wave speed. The below figures are a recreation representative of a receiver's data for two pulses. The blue dotted curve represents the intensity Vs time curve for the reception of the 310c speed pulse and the red curve represents the intensity Vs time curve for the arrival of a c speed pulse sent at the same time.
One can see on the close up second graph from the horizontal shift that the 310c curve arrived 62ns earlier than the c speed pulse. The question then arises whether this experiment is an example of a causality violation. In order for causality to be violated one must have faster than c "information" transfer. Under typical "long" transmissions one can consider the information transfer speed to be the group wave speed or the speed of the energy carried by the pulse. It has long been known that phase wave speeds often exceed c which is why it is often pointed out that the energy transmission in ordinary wave-guides occurs at the group speed which is less than c. As such, the information transfer speed is less than c in ordinary wave guides. The reason that the group speed exceeding c for this experiment is not convincing of faster than c information transfer and the reason why the information transfer for this experiment can not be taken to be the group speed is because of the following. The 62ns time shift between the two pulses is much less than the time it takes to receive the entirety of a pulse itself. Thus the time it took from the time the pulse began to enter the medium until the time the receiver read the entirety of the information was the sum of group transfer and read times. The full width at half-max FWHM of the pulse in question is approximately . Take that to be read time. The group transfer time was
6.0cm/310c = 0.65ps. Taking the information transfer time to be the sum of these, approximately just , one finds that the information speed was
, a mere measly fraction of the vacuum speed of light. There are two ways one might modify this experiment so that if successful it would clearly demonstrate faster than c information transfer. First, one might make the medium of transfer much longer so that in the information transfer time it is the read time that is insignificant instead. The reason that this may be an impossible task is that due to the dispersive nature of the media itself, even with the gain assistance, there will be a trade off between the signal degradation and length. There may be a limiting trade off so that in a long enough transmission line so that the information transfer time yields a faster than c speed the signal would have been lost. Second, one might try to significantly narrow the pulse so that the information transfer time is approximately just the group transfer time. The reason that this may be an impossible task two fold. There is a narrow frequency range at which the light must be sent through the medium in order for it to transfer with a faster than c group speed. This in itself puts a limit on how narrow the pulse may be. Also, the narrower the pulse is made the more rapidly it will tend to widen itself as it travels across the medium. By the time it gets to the other end the read time will always be longer than the send time so no matter how short the send time is made one will have to contend with a longer read time.
Though such experiments do successfully demonstrate faster than c group transfer, they do not conclusively demonstrate the faster than c information transfer, which they would have to in order to show a causality violation.
A particle that travels less than the Lorentz invariant speed c is a tardyon. A particle that travels at the Lorentz speed c in true vacuum is a luxon. And a particle that travels faster than the Lorentz speed is a tachyon. Most particles so far experimentally observed have been tardyons. As far as experiment has been able to determine photons have zero mass. As such if they can ever be said to travel in a true vacuum they would do so at the speed c and so photons are luxons. There are three experimentally observed particles that are candidates for the title of tachyon, the three flavors of neutrino, the electron neutrino, the muon neutrino and the tau neutrino.
CERN once announced that they had had positive results for the neutrino traveling as a tachyon. What they did is smash atoms at a collider at CERN which emits a pulse of neutrinos. They detect the neutrinos at the OPERA neutrino detector at San Grasso and then divide the distance between them by the time between the creation and detection events to get the speed of the neutrinos. The claim was that the neutrinos arrived there 60 billionths of a second sooner than light in vacuum could transverse the same distance. They said their system's measurement errors were within 10 billionths of a second. Their claim essentially was in essense that they had two clocks 732 kilometers apart that they were keeping in sync to within 10 billionths of a second despite the distance using GPS, and that they knew the time of the creation event and the time of the detection event to within as much error. A problem with their finding was found in that the synchronization of the clocks was according to local GPS satellites which are in motion with respect to the labs. If a special relativistic length contracted standard of distance is considered in determining the locations of the events this 60 ns discrepancy is completely accounted for within the orthodox application of special relativity. See http://arxiv.org/abs/1110.2685
I think as a publicity stunt, they went on with the media about how shocked they were to stumble onto this result, getting a faster than c answer. In reality they did the experiment full well knowing that this might be the answer they would get and that was motive for doing the experiment in the first place. There have been tachyon neutrino models in the fringe for over a decade now whose predictions regarding neutrino chirality does match observation. Also even before this experiment neutrino arrival from supernovae have been compared to the times we have been seeing them optically with positive results. Typically neutrino detectors will get a pulse of neutrinos a few hours before a supernova close enough to see will be spotted. The reason this hasn't been taken as proof of tachyons is because the light moving through the plasma of the star will be slowed down a few hours due to the index of refraction so the difference isn't significant enough to be sure. So, if neutrinos are tachyons the ones emitted from supernovae barely travel faster than light in vacuum at all.
The speed CERN originally got though nearly c is significantly different enough that should neutrinos from supernovae travel at the CERN calculated speed, they should arrive here years, not mere hours before we see the supernova. It is possible that tachyons from supernovae are much more energetic causing them to travel slower, closer to the speed of light. Yeah that sounds weird, but that's how tachyon's physics should work.
But to demonstrate this discrepancy with supernova neutrino speeds consider the example calculation:
The difference between v/c or for the neutrinos and 1 reported for their experiment was
The difference in time for the arrival of the neutrinos and the arrival of a light speed signal in vacuum given this speed for neutrinos should be
A little algebra yeilds
where t is the time since the neutrinos and light speed particles were created. Since is small this is approximately
Supernova SN 1987A occurred 163,000 years ago. So if the neutrinos were to travel at the CERN calculated speed as compared to the light they should have arrived 4.04 years before the light did. However the light arrived a mere 3 hours after the neutrinos. Since the 3 hours can even be completely accounted for by the light slowing passing through the stellar matter that the neutrinos are not impeded by this casts great doubt on the premise that neutrinos ever travel faster than c at all.
There is another possible contribution to such a discrepancy that would also need to be ruled out had the results been repeated by other groups. Space is not a true vacuum. As near as it is, with all the radiation it contains and with the presence of quantum foam, space may have some slightly different index of refraction than a true vacuum would have. If that is the case then light will travel some small amount less than the true value of c through space while neutrinos can travel closer to the true value not effected by the index of refraction. We would then measure their speed as if it were faster than light even though they may be traveling less than or equal to the true value of c.
Aside from throwing our certainty of the principle of causality into doubt and opening up the possibility of time travel as has been discussed, the theoretical special relativistic physics of tachyons really isn't too different from tardyons. The group speed of a real particle is
and the phase speed is
So the relation between group and phase speed for a real particle is
So no matter what kind of particle you have the energy momentum group and phase velocities can all be real and the primary difference is that for tachyons the group speed is greater than c while the phase speed is less than c whereas for tardyons the group speed is less than c while the phase speed is greater than c.
In the end of all the drama, a second group ran the experiment and did not get the faster than c result at all. Though I think the truth was too embarrassing to admit that they had left out the length contraction on lab events or equivalently the relative simultaneity between labs Vs GPS satellite frame, in the end they blamed the initial faster than c measure on a loose cable!
Consider monochromatic plane waves of light with a wavelength of 500nm. What is the frequency of the light?
a. How much kinetic energy does a 75 kg man have if he is traveling to another star at (4/5)c ?
b. Though "system mass" i.e. center of momentum frame energy is conserved in an isolated reaction, the sum of masses is not. The sum of kinetic energies of a system's parts after a reaction is the sum of kinetic energies of the system's parts before the reaction plus the difference in the sum of masses that the system decreased by through the reaction. Changes in binding energies of interacting particles in the system are typically accounted for in the sum of masses. For example a hydrogen atom in a system is least massive in its ground state. So, such a reaction is sometimes called a "mass to kinetic" energy conversion. How much of the sum of masses of a system would have to be annihilated in order to produce the same amount of kinetic energy as the man had in part a? (An atomic bomb "converts" about the mass of a coin to kinetic energy in its explosion)
Show that if the speed of an object is where is an extremely small number for a high speed, that can be approximated by
(Modified from Misner, Thorne, and Wheeler's Gravitation problem 5.4)
Consider a system with .
a. Use the Lorentz transformation property of a tensor to show that
Note- is defined in equation 3.1.16 for general stress-energy tensors.
b. Derive the equations of part a. from Newtonian considerations plus the conservation of relativistic energy. (Hint: The total energy carried past the observer by a volume of the medium V includes both the rest energy T ' 00 V and the work done by forces acting across the volume's face as they push the volume of the medium through a distance.)
c. Refer to equation 3.1.26 a. Considering that force is a change in momentum with respect to time, this results in a force per unit volume f n of
f n = dT 0n/dct
Which for constant i n k results in
This equation suggests that one call an inertia density matrix. What does this become for the special case of a perfect fluid?
d. Consider an isolated, stressed body at rest and in equilibrium according to the laboratory frame. Show that its total inertia matrix I n k (not moment of intertia) defined by
is isotropic and results from the mass or center of momentum energy of the body
Show that if
For special relativity, what is the speed, frequency, and wavelength of a 2Mev proton?
Use equation 3.1.4b to find the constant in equation 3.1.2d.
For a particle in a four vector potential , the mass will be defined as the following invariant m in:
where q is the charge reacting to that field.
In special relativity this results in
If one were to replace the energy and momentum with their operators and operate this on a wave function the result would be second order differential equations. In order to get a first order differential equation one could take the square root of the equation if the right hand side could be expressed as a perfect square. As it stands one can not do this, but consider the expression multiplied by the 2x2 identity matrix I of 2x2 identity matrices .
Now one can hope to find a perfect square for the right hand side in the form
where the three elements of F are Fi, and where Fi and G are 2x2 martices of 2x2 matrices.
where si is any one of the three conventional Pauli matrices and verify that this yields
This results in a hamiltonian of
(often multiplied identity matrices are suppressed and identity by itself written as just 1)
and the Dirac equation for a nonzero potential:
3.2 The Special Relativity Dynamical Equations 27
In special relativity theory we define a four-vector force as
For a particle with mass in terms of the velocity 4-vector, we have
In SR, the Acceleration Four-Vector is given from the velocity 4-vector by
and so in SR we can write the relativistic version of Newton's second law as
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Considering an inertial frame according to which the test mass is instantaneously at rest it is easy to show that
But this product will yield an invariant to Lorentz transformations, so this serves as the work energy theorem for the physics of special relativity theory as shown:
Consider where leads:
If we define another kind of force that is not a four-vector which we will call ordinary force as
f = dp/dt
then we arrive at
and there we see the work energy theorem of pre-modern relativistic physics.
So Newton's second law in terms of ordinary force is 3.2.6
f i = dp i/dt
where p i is the relativistic 3-momentum. Using this force definition has its purposes, but in a lot of ways thinking of relativistic physics in terms of nontensor quantities very much complicates things. For example let us work out the relation between ordinary force and coordinate acceleration.
we can write 3.2.6 as
We then define by
Which for acceleration in the direction of motion results in
and for that case of motion will be equal to the proper acceleration, A', which is the acceleration as observed from a frame according to which the particle is instantaneously at rest. The magnitude of this can be calculated from any frame as it is an invariant, . This is the amount of acceleration "felt" by the accelerated observer and according to the inertial frame in which the accelerated or "proper frame" observer is instantaneously at rest a = A'. We define according to 3.2.8 as well because it is useful. When is calculated from any inertial frame according to which the force is in the direction of the motion, it will then turn out to be equal to the proper acceleration, A'.)
This also restores a Newtonian form
( Note also - When the force is in the same direction as the motion, then the force felt by the object being pushed is equal to the ordinary force. In that case we have )
3.2 The Special Relativity Dynamical Equations 29
we can eliminate , in terms of dt from time dilation
Use of the chain rule and simplification results in
where is the coordinate 3-acceleration.
The four-vector force for the physics of Einstein's special relativity theory is sometimes called the Minkowski force and is related to the electromagnetic field tensor
From this we can work out the relation between the components of the electromagnetic field, the coordinate velocity and the ordinary force, which yields
In the case that the force is in the direction of the motion 3.2.10 yields,
Note that no matter what finite value the ordinary force is, as u approaches c, diverges and so the acceleration must vanish.
We expect this as nothing with mass can be pushed all the way up to the speed of light.
We have seen how c is a speed limit for the universe. Because of this, we must answer a question concerning velocity addition. Lets say an S' frame observer observes an object at speed u'. An S
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frame observer observes the S' observer to be moving at speed v. u' can be any value less than c and v can be any value less than c. People tend to come to the wrong conclusion that the S observer observes that the object moves at a speed u = u' + v and that this speed should therefor be any speed less than 2c. They are using the wrong velocity composition formula. Consider the following Lorentz coordinate transformation equations in differential form.
To obtain the correct velocity addition equation divide equations and simplify.
Now making the replacements ux = dx/dt and u'x = dx'dt' we arrive at
This is the correct equation to use for that velocity addition. u' and v can be any values less than c but the result will always be that the speed of the object according to S, u , will always be less than c. One can also use the same method to find the velocity addition equations for the case that the object moves in the y or z directions.
If one allows the direction of velocity of the primed observer with respect to the unprimed observer to be arbitrary and lets the direction of the velocity of the object being observed also to be arbitrary then the velocity addition formula from equation 1.1.3 the Lorentz transformation equations for arbitrary direction is
As an example lets see how 3.2.12 comes out of 3.2.13 for the case that
In this case 3.2.13 yields
Next lets see what we get for the other components for this case of velocity between the frames:
Symmetrically you get the same relation for the z components.
Next lets show that 3.2.13 yields an invariance of the speed c for arbitrary directions involved.
Start by taking the inner product of 3.2.13 with itself, then simplify
Now the final equation is important
If |u'| = c then this reduces to |u| = c and this is true regardless of the velocity of the frames, regardless of the direction of the velocity of the frames and regardless of the direction of the c speed thing being observed. As such the Lorentz transformation equations for arbitrary direction given by equation 1.1.3 do indeed yield an invariant speed to all frames related by that transformation.
Find , , and , using the fact that the full contraction of any tensor is an invariant. How is the last related to the definition of mass? How is the result for related to the time derivative of the work energy relation, ?
Show that the momentum for a charged particle in cyclotron motion or orbital radius R in a magnetic field is given for special relativity by p = qBR.
Use Eqn 3.2.10 to show that the motion of a charged particle in an electromagnetic field is given by
Refer to Eqn 3.2.2 and show that
Consider an S' frame moving in the x direction with speed v with respect to S.
Show that the ordinary force on a particle moving with velocity u with respect to S transforms between the two according to
This is not a tensor transformation.
Consider the ordinary force as a columb vector
and the and the coordinate acceleration given by the columb vector
Show that according to equation 3.2.10, these can be related by
if we define *matrix mass* Mij by
Sine one does not ordinarily wish to think of mass as a matrix it is generally not good to think of mass as the ratio of ordinary force to coordinate acceleration. Instead it is the ratio of a given element of the four-vector force to the corresponding element of four-vector acceleration which is a simple invariant scalar and carries with it the meaning of resistance to deviation from geodesic motion instead of resistance to change in motion as it means in Newtonian dynamics.
Take the inner product of 3.2.13 with itself and use some algebra to show that if |u'| = c then it is also the case that |u| = c regardless of what v is or the directions involved.
For the figure lets say that the velocity of ship B with respect to planet A is (1/2)c after departure in the +x direction and the speed of ship C with respect to planet A is (1/2)c after departure in the -x direction. There are triplets one of which is on planet A, one of which is on ship B and one of which is on ship C. The departure happened on their 30th birthday. After departure they remain in states of constant velocity forever.
a. According to the standard of simultaneity for the one on planet A, show that the ages of the siblings on A's 35th birthday are 34.3 years old.
b. How fast is ship B moving with respect to ship C?
c. According to the standard of simultaneity for the one on ship C how old are the siblings when C is 34.3 years old?
d. According to the standard of simultaneity for the one on ship B how old are the siblings when B is 34.3 years old?
3.3 Rotations, Rockets, and Frequency Shifts 31
We have shown that velocities do not add linearly in the physics of Einstein's special relativity theory. For motion along one direction velocities were adding nonlinearly according to Eqn 3.2.12
Rapidity as a function of v is given by
This definition is useful as it simplifies much of dynamics equations. It does this because, unlike velocity, considering motion along the x direction, rapidity does add linearly.
Recall the Lorentz transformation matrix
Rapidity also has the following relations to and
From these, the Lorentz transformation matrix becomes
Comparing this to an ordinary rotation matrix makes it clear why Lorentz transformations can be thought of as a rotation in space-time. At this point the relation between Lorentz transformation and rotation may still seem to be a superficial one, but once one becomes familiar with spinor calculus a much more intimate relation is revealed.
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Here we will derive and discuss the implications of single stage relativistic rocket equations. The non-relativistic rocket equation is
This gives the change in velocity a rocket undergoes accelerating in one direction given a measure of exhaust speed vex which is a constant and the initial mass of the rocket mi and the final mass of the rocket m after some of the ships mass in fuel has been burnt off.
The relativistic version of this equation in terms of rapidity is similar
The speed of the rocket is then calculated from the rapidity Eqn 3.3.1.
Notice that since for any , v is always less than c no matter how much of the ships mass is burnt off as fuel and no matter how fast the exhaust speed is. We can even consider tachyon exhaust where and yet the rocket still never reaches the speed of light.
To derive 3.3.4 Start with conservation of momentum and energy relating the initial and final states of the rocket and exhaust for a small element mfex burned off.
After integration equation 3.3.4 is obtained
Now consider the ships proper acceleration for motion in one direction refer to equation 3.2.8b
If the proper acceleration is kept constant then integration results in
Consider initial conditions of v = 0 at t = t' = 0.
If the rocket starts at rest and is run at a constant proper acceleration , then the equation can be written
These initial conditions also result in
Inverting these results in
Running it at a constant proper acceleration also results in
3.3 Rotations, Rockets, and Frequency Shifts 33
Using the Lorentz like transformation equations
Results in a good global coordinate transformation from the accelerated ship frame to an inertial frame. Let it start instantaneously at rest with respect to the latter at t' = 0 and these become
There is a difference between what frequency one observes as being emitted from a source and what frequency an observer actually sees as coming from the source. This is true even in non-relativistic physics. For instance, as a car drives past you will hear a shift in the tone of the engine as it goes from coming toward you to going away from you. This is the frequency you hear. You may use the ordinary Doppler shift formula with the speed it was traveling to then extrapolate what frequency it really emits according to your coordinate frame. This is the frequency you observe. The relativistic Doppler shift formula is really the same thing as the ordinary Doppler shift formula except that it is usually written in terms of the source frame's emitted frequency instead of the observed emitted frequency. Its just that in the non-relativistic case these are the same. In relativistic Doppler shift, you accounts for the fact that due to time dilation the frequency you observe to be emitted is different then the frequency according to the frame of the object.
If you are at rest with respect to the medium of propagation for a wave, then the ordinary Doppler shift formula is
In terms of sound we make the following relations.
f is the frequency you hear (for instance if this was sound).
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f0 is the frequency you observe have been emitted according to your frame at the time of the emission. It is the transverse or frequency for f.
v is the speed the emitter travels with respect to the medium of the waves at the time the wave was actually emitted.
c is the speed of the waves of the medium with respect to the medium.
is the angle it was traveling off of strait away from you at the time the heard frequency was actually emitted.
Relativistic Doppler shift IS ordinary Doppler shift. This formula happens to stand correct for the relativistic Doppler shift of light with the following adjustments to the relations.
f is the frequency you see.
f0 is the frequency you observe to have been emitted according to your coordinates at the time of the emission. It is the transverse or frequency for f.
v is the speed the emitter travels with respect your frame at the time the light was actually emitted.
c is the Lorentz invariant vacuum speed of light.
is the angle it was traveling off of strait away from you at the time the heard frequency was actually emitted according to your frame. <--- The angle is different according to the other frame and so use of the other frames angle changes the form of the equation.
Now to write it in terms of the frequency emitted according to the frame of the object we start by writing the periods according to the two frames in terms of time dilation.
We then relate frequency to period.
f0 = 1/T
f0' = 1/T0'
3.3 Rotations, Rockets, and Frequency Shifts 35
Putting these together results in
f0 = f0' (1 - v2/c2)1/2
Inserting this into the Doppler shift formula results in
The wavelength of the light will be , or...
Next consider the case that the object travels strait toward the observer. . Then after algebraic simplification these becomes
If the object traveled strait away from the observer , it would have become
Consider an ideal matter/antimatter rocket with vex = c. What is the mass ratio mi/m in order to reach (4/5)c? Even if it were technologically possible to make that much, think about how much anti-matter becomes a serious safety concern.
Consider an ideal matter/antimatter rocket with vex = c. What is the mass ratio in order to accomplish the round trip of problem 1.2.4 ?
What wavelength do you see from a 557nm proper frame source approaching at (5/13)c?
a. Show that 3.3.6a yields 3.3.6b
b. Show that inverting 3.3.6b yields 3.3.6d
Show that the relativistic Doppler formula for a source traveling toward an observer equation 3.3.13 can be written as
v = ctanh[ln(f/f0')]
f/f0' = exp[tanh-1(v/c)]
and the case of the source traveling away from the observer 3.3.14 can be written
v = ctanh[ln(f0'/f)]
f0'/f = exp[tanh-1(v/c)]
With these forms one can punch numbers easier on a scientific calculator.
Show that the position of the space vehicle undergoing constant proper acceleration considered in equations 3.3.9 is given as a function of proper time by
a. Use 3.3.9 to show that the wrist watch time or proper time for the ship frame observer describing events at the location of the ship is related to coordinate time t by
b. Use the results of "a" and 3.3.5 to show that
and look at the limiting behavior of v for both small and large times for both the velocity equation here and 3.3.5.
c. Show that the position of the ship undergoing constant proper acceleration considered in equations 3.3.9 is given as a function of proper time by
and use the results of "a" to show that
Note that this is the equation of a hyperbola.
A space ship of constant mass is somehow propelled by an external force. The ordinary force on the ship will be constant and will be in the direction of motion so one may use 3.2.11. Use this and 3.2.9 to show that
and use this result to show that
Then compare the results to problem 3.3.6.
Bergmann, P. G., 1942, Introduction to the Theory of R., Prentice-Hall, New York
Birkoff, G. D., 1923 R. and Modern Physics, Harvard University Press, Cambridge, Mass.
Hagedorn, R., 1964, Relativistic Kinematics, W. A. Benjamin, New York
Infeld, L., and J. Plebanski, 1960, Motion and R., Pergamon, New York