RELATIVITY OF MASS-really interesting
hen something is moving past extremely fast, its appearance changes - a full analysis shows the object as rotated and forshortened. The forshortening is easily shown and is called RELATIVISTIC LENGTH as opposed to its proper length.
Let us go back to our alien who is now using light and a stopwatch to measure the speed v of his ship relative to Earth using signals from Earth. He knows the Proper Length l0 of his ship but he will be using a relative time interval of ΔT as he will not be judging the passage of his ship. He will be relying on flares from us.
WE, on Earth, will measure the same speed v as the spaceship passes and signal with flares when we judge the front and rear exactly in front of us. We will get a proper time interval ΔT0 as we are judging the event, but any length of the ship arising will be relativistic!
v = l / ΔT0 = l0 /ΔT v = speed of ship relative to Earth, l = relativistic length
Now use the earlier results relating relativistic time to proper time and we get
RELATIVISTIC LENGTH < PROPER LENGTH
MASS
Special Relativity assumes both Conservation of Momentum & Conservation of Energy. These are much more fundamental assumptions than Newton's Laws which can be derived from these conservation principles at low speeds.
If a rocket has a motor going flat out for a very long time, relative to Earth, it will acquire a lot of kinetic energy, but not an infinite kinetic energy as the rocket will NEVER exceed the speed of light relative to Earth. ( Light will always pass it at the speed of light! )
If we now collide this very fast rocket with something to measure its momentum and energy - and assume these laws are conserved, we find the MASS >> than its proper or "rest" mass!
Rough derivation.
Suppose our alien chucks a ball from his fast moving ship which happens to be identical to a ball ( mass m0 ) we throw. We watch the subsequent collision which, by pure chance takes place after both balls move a distance d at right angles ( transverse ) to the ship's motion as measured by us.
The experiment is such that we believe that the two transverse momenta cancel - ie the balls' momenta are exactly equal and opposite of each other.
We see a
PROPER measurement of our ball's momentum - p0 = m0 d / ΔT0
and a RELATIVISTIC measurement of the alien ball's transverse momentum p = m d / ΔT m = relativistic mass
But the collision has it that the TOTAL momentum = 0 in this direction, so m0 d / ΔT0= m d / ΔT
Immediately m0 / ΔT0= m / ΔT so
As the relative speed increases, the mass appears to increase. The RELATIVISTIC MASS increases.
The Proper Mass is also called the REST MASS. This is the mass you measure on an ordinary balance. After all, measuring mass IS an experiment you have carried out in your frame of reference - your lab!
Example; An electron has a rest mass of 9.11 x 10-31 kg. What is its relativistic mass when travelling at 0.9c in an old TV tube?
Soln. m = m0 / γ = 9.11 x 10-31 / ( 1 - 0.81 )1/2 = 2.1 x 10-30 kg
In actual fact, most old TV tubes operate at 28kV giving a speed of close to 0.33c, what will the relativistic mass be in these situations?
When it collides with the screen, it has this mass for the purposes of calculating the collision.
MASS-ENERGY
If nothing can go infinitely fast but is limited always to be below the speed of light, what happens when we add energy to our moving object? The relativistic mass increases - the added energy becomes relativistic MASS-ENERGY. Indeed, the Rest Mass is also energy, the rest energy. Mass and energy become interchangeable in the most famous of Einstein's equations.
E = mc2 , E is the TOTAL energy of the particle, m is the relativistic mass.
Kinetic energy is now the difference between the total energy and the rest mass energy
Ek = mc2- m0c2
The implications are profound.
ALL chemical reactions in which energy is released or absorbed, MASS CHANGES. This is not what chemists would have you believe.
Suppose a power station buring coal produces 200MW. Then 200MJ a second = 200 x 106 / (3 x 108)2 kg a second. Each day this amounts to 1.92 x 10-4 kg . If we store all the products and get their mass very precisely and compare with a very precise measurements of the reactants we would get a deficit of 1.92 x 10-4 kg each day.
It is tiny - chemists quite rightly ignore it - but it is as real as in nuclear reactions. A 200MW nuclear reactor will show EXACTLY the same total mass loss each day.
Light has mass-energy!
Photons have energy given by E = hf , Planck's equation. Compton equated this with E = mc2 and gave photons a momenta of
p = mc = hf / c = h / λ so a "mass" is assigned m = hf / c2 . This makes light susceptible to gravity, the beginning of General Relativity.
eg; A green photon of wavelength 470 nm will have a momentum of 1.41 x 10-27 kgms-1, and an energy mass of 4.7 x 10-36 kg. Obviously light has no rest mass!
( Note that momentum for light can be described through Maxwell's Equations as well. This value is that given by a very large number of photons. )
RELATIVE VELOCITIES
This takes us right back to the beginning - how do the aliens passing each other at 0.9c ( according to us ) see each other? They see us pass at 0.9c - no problems. But if I am in one of the alien spaceships?
The formula is quite messy. For one direction ONLY - no fancy 2D stuff of the opening page,
( If we work in 2D, it gets worse. You need a second set of different equations for the y axis. )
Notice the extra term in the denominator which guarantees the relative speed will be less than the speed of light. For low speeds this term is clearly close to zero.
Substituting for the 0.9c for each AND that they are in opposite directions gives
vAlienB rel A = [0.9c - (-0.9c)] / [1 - (-0.9c)*0.9c/c2] = 1.8c /1.81 = 0.994c NOT 1.8c
Advanced animations showing relativistic effects created at Australian National University.
Problems
1. a) If an alien passes us at 0.99c and measures a metre ruler sitting on the ground next to us, what length will it get? ( 0.14m )
b) What time will a light pulse take to travel the length of the ruler according to us? (3.33 x 10-9 s )
c) What time will the light take to travel the length of the metre ruler according to the alien? ( 4.67 x 10-10 s )
2.. You are to travel at a constant velocity of 0.97c from Earth to Alpha Centaurus, 4.26 light years away.
There will be MANY other hazards trying this.
3. What is the rest mass energy of a proton? ( 1.5 x 10-10 J or 9.4 x 108 eV )
4. A proton is now accelerated to 0.99c relative to us. What is its relativistic mass? What is its mass energy in eV? Where might protons acquire such mass energy? ( 1.18 x 10-26 kg, 6.66 GeV )
5. Cosmic rays are mixtures of charged particles and Gamma Rays. Extremely energetic gamma rays occasionally occur which have the energy of a thrown ball - even over 1020 eV! Current physics cannot readily explain them.
a) What is 1020 eV in J ? What speed would a 0.1kg ball be travelling at to attain this energy? ( 16J, 17.9 ms-1 )
b) What photonic frequency does this correspond to? And what photonic mass- energy in kg does this correspond to? ( 2.7 x 1034Hz, 1.78 x 10-16kg )
c) If a proton has a mass energy of 1020 eV , with what fraction of the speed of light is it moving? (0.999999999999999999999956c)
6. CERN is currently building the Large Hadron Collider of energy 14 TeV, for protons. What fraction of the speed of light will these particles be moving at if each is at half this value? ( The LHC takes particles travelling in opposite directions with equal energy to give the total energy. ) ( 7 TeV protons are travelling at 0.999999991c )
7. For a PET scanner, an artificial radioactive source such as C-11 emits an antielectron. The antielectron annihilates an electron and two gamma rays travelling in opposite directions are produced. From the mass of the electron ( 9.11 x 10-31 kg) and the identical mass for the antielectron, calculate the energies of each of the gamma rays in MeV. What frequency does this correspond to? ( 0.511MeV, 1.2 x 1020 Hz )
8. What energy must a proton acquire to create an electron? ( Other factors also enter making this a naive calculation.) ( 0.511MeV )
9. a) You, on Earth, observe Superman passing at 0.7c but Superwoman passes from the opposite direction at 0.8c. What speed does Superman see Superwoman pass at? ( 0.968c )
b)What if Superwoman is trying to overtake Superman with the same speeds as above but in the same direction? What will Superman now see? ( 0.23c )
PS - my thanks to Craig Savage of Australian National University for his comments and corrections of some of these relativity pages where I got it wrong!
hen something is moving past extremely fast, its appearance changes - a full analysis shows the object as rotated and forshortened. The forshortening is easily shown and is called RELATIVISTIC LENGTH as opposed to its proper length.
Let us go back to our alien who is now using light and a stopwatch to measure the speed v of his ship relative to Earth using signals from Earth. He knows the Proper Length l0 of his ship but he will be using a relative time interval of ΔT as he will not be judging the passage of his ship. He will be relying on flares from us.
WE, on Earth, will measure the same speed v as the spaceship passes and signal with flares when we judge the front and rear exactly in front of us. We will get a proper time interval ΔT0 as we are judging the event, but any length of the ship arising will be relativistic!
v = l / ΔT0 = l0 /ΔT v = speed of ship relative to Earth, l = relativistic length
Now use the earlier results relating relativistic time to proper time and we get
RELATIVISTIC LENGTH < PROPER LENGTH
MASS
Special Relativity assumes both Conservation of Momentum & Conservation of Energy. These are much more fundamental assumptions than Newton's Laws which can be derived from these conservation principles at low speeds.
If a rocket has a motor going flat out for a very long time, relative to Earth, it will acquire a lot of kinetic energy, but not an infinite kinetic energy as the rocket will NEVER exceed the speed of light relative to Earth. ( Light will always pass it at the speed of light! )
If we now collide this very fast rocket with something to measure its momentum and energy - and assume these laws are conserved, we find the MASS >> than its proper or "rest" mass!
Rough derivation.
Suppose our alien chucks a ball from his fast moving ship which happens to be identical to a ball ( mass m0 ) we throw. We watch the subsequent collision which, by pure chance takes place after both balls move a distance d at right angles ( transverse ) to the ship's motion as measured by us.
The experiment is such that we believe that the two transverse momenta cancel - ie the balls' momenta are exactly equal and opposite of each other.
We see a
PROPER measurement of our ball's momentum - p0 = m0 d / ΔT0
and a RELATIVISTIC measurement of the alien ball's transverse momentum p = m d / ΔT m = relativistic mass
But the collision has it that the TOTAL momentum = 0 in this direction, so m0 d / ΔT0= m d / ΔT
Immediately m0 / ΔT0= m / ΔT so
As the relative speed increases, the mass appears to increase. The RELATIVISTIC MASS increases.
The Proper Mass is also called the REST MASS. This is the mass you measure on an ordinary balance. After all, measuring mass IS an experiment you have carried out in your frame of reference - your lab!
Example; An electron has a rest mass of 9.11 x 10-31 kg. What is its relativistic mass when travelling at 0.9c in an old TV tube?
Soln. m = m0 / γ = 9.11 x 10-31 / ( 1 - 0.81 )1/2 = 2.1 x 10-30 kg
In actual fact, most old TV tubes operate at 28kV giving a speed of close to 0.33c, what will the relativistic mass be in these situations?
When it collides with the screen, it has this mass for the purposes of calculating the collision.
MASS-ENERGY
If nothing can go infinitely fast but is limited always to be below the speed of light, what happens when we add energy to our moving object? The relativistic mass increases - the added energy becomes relativistic MASS-ENERGY. Indeed, the Rest Mass is also energy, the rest energy. Mass and energy become interchangeable in the most famous of Einstein's equations.
E = mc2 , E is the TOTAL energy of the particle, m is the relativistic mass.
Kinetic energy is now the difference between the total energy and the rest mass energy
Ek = mc2- m0c2
The implications are profound.
ALL chemical reactions in which energy is released or absorbed, MASS CHANGES. This is not what chemists would have you believe.
Suppose a power station buring coal produces 200MW. Then 200MJ a second = 200 x 106 / (3 x 108)2 kg a second. Each day this amounts to 1.92 x 10-4 kg . If we store all the products and get their mass very precisely and compare with a very precise measurements of the reactants we would get a deficit of 1.92 x 10-4 kg each day.
It is tiny - chemists quite rightly ignore it - but it is as real as in nuclear reactions. A 200MW nuclear reactor will show EXACTLY the same total mass loss each day.
Light has mass-energy!
Photons have energy given by E = hf , Planck's equation. Compton equated this with E = mc2 and gave photons a momenta of
p = mc = hf / c = h / λ so a "mass" is assigned m = hf / c2 . This makes light susceptible to gravity, the beginning of General Relativity.
eg; A green photon of wavelength 470 nm will have a momentum of 1.41 x 10-27 kgms-1, and an energy mass of 4.7 x 10-36 kg. Obviously light has no rest mass!
( Note that momentum for light can be described through Maxwell's Equations as well. This value is that given by a very large number of photons. )
RELATIVE VELOCITIES
This takes us right back to the beginning - how do the aliens passing each other at 0.9c ( according to us ) see each other? They see us pass at 0.9c - no problems. But if I am in one of the alien spaceships?
The formula is quite messy. For one direction ONLY - no fancy 2D stuff of the opening page,
( If we work in 2D, it gets worse. You need a second set of different equations for the y axis. )
Notice the extra term in the denominator which guarantees the relative speed will be less than the speed of light. For low speeds this term is clearly close to zero.
Substituting for the 0.9c for each AND that they are in opposite directions gives
vAlienB rel A = [0.9c - (-0.9c)] / [1 - (-0.9c)*0.9c/c2] = 1.8c /1.81 = 0.994c NOT 1.8c
Advanced animations showing relativistic effects created at Australian National University.
Problems
1. a) If an alien passes us at 0.99c and measures a metre ruler sitting on the ground next to us, what length will it get? ( 0.14m )
b) What time will a light pulse take to travel the length of the ruler according to us? (3.33 x 10-9 s )
c) What time will the light take to travel the length of the metre ruler according to the alien? ( 4.67 x 10-10 s )
2.. You are to travel at a constant velocity of 0.97c from Earth to Alpha Centaurus, 4.26 light years away.
a)What is the time to travel the distance from Earth's perspective? What is your time for the same trip? ( 4.39y, 1.07y )If you are to travel to Alpha Centaurus, you must accelerate then decelerate. The calculations above do not take this into account. Accelerational effects are quite complex on time.
b) In what sense does the results of 1. above make interstellar travel highly unlikely?
c) Recalculate the spaceship time in 1. for 0.999c velocity. ( 0.19y )
There will be MANY other hazards trying this.
3. What is the rest mass energy of a proton? ( 1.5 x 10-10 J or 9.4 x 108 eV )
4. A proton is now accelerated to 0.99c relative to us. What is its relativistic mass? What is its mass energy in eV? Where might protons acquire such mass energy? ( 1.18 x 10-26 kg, 6.66 GeV )
5. Cosmic rays are mixtures of charged particles and Gamma Rays. Extremely energetic gamma rays occasionally occur which have the energy of a thrown ball - even over 1020 eV! Current physics cannot readily explain them.
a) What is 1020 eV in J ? What speed would a 0.1kg ball be travelling at to attain this energy? ( 16J, 17.9 ms-1 )
b) What photonic frequency does this correspond to? And what photonic mass- energy in kg does this correspond to? ( 2.7 x 1034Hz, 1.78 x 10-16kg )
c) If a proton has a mass energy of 1020 eV , with what fraction of the speed of light is it moving? (0.999999999999999999999956c)
6. CERN is currently building the Large Hadron Collider of energy 14 TeV, for protons. What fraction of the speed of light will these particles be moving at if each is at half this value? ( The LHC takes particles travelling in opposite directions with equal energy to give the total energy. ) ( 7 TeV protons are travelling at 0.999999991c )
7. For a PET scanner, an artificial radioactive source such as C-11 emits an antielectron. The antielectron annihilates an electron and two gamma rays travelling in opposite directions are produced. From the mass of the electron ( 9.11 x 10-31 kg) and the identical mass for the antielectron, calculate the energies of each of the gamma rays in MeV. What frequency does this correspond to? ( 0.511MeV, 1.2 x 1020 Hz )
8. What energy must a proton acquire to create an electron? ( Other factors also enter making this a naive calculation.) ( 0.511MeV )
9. a) You, on Earth, observe Superman passing at 0.7c but Superwoman passes from the opposite direction at 0.8c. What speed does Superman see Superwoman pass at? ( 0.968c )
b)What if Superwoman is trying to overtake Superman with the same speeds as above but in the same direction? What will Superman now see? ( 0.23c )
PS - my thanks to Craig Savage of Australian National University for his comments and corrections of some of these relativity pages where I got it wrong!
