Chapter 26Chapter 26RelativityRelativity

General Physics

Relativity IIRelativity II

Sections 5–7Sections 5–7

General Physics

Relativistic DefinitionsRelativistic Definitions

To properly describe the motion of To properly describe the motion of particles within special relativity, Newton’s particles within special relativity, Newton’s laws of motion and the definitions of laws of motion and the definitions of momentum and energy need to be momentum and energy need to be generalizedgeneralized

These generalized definitions reduce to These generalized definitions reduce to the classical ones when the speed is the classical ones when the speed is much less than cmuch less than c

General Physics

Relativistic MomentumRelativistic Momentum

To account for conservation of momentum in all To account for conservation of momentum in all inertial frames, the definition must be modifiedinertial frames, the definition must be modified

vv is the speed of the particle, is the speed of the particle, mm is its mass as is its mass as measured by an observer at rest with respect to the measured by an observer at rest with respect to the massmass

When v << c, the denominator approaches 1 and so When v << c, the denominator approaches 1 and so pp approaches approaches mvmv

2 21

mvp mv

v c

General Physics

Relativistic CorrectionsRelativistic Corrections

Remember, relativistic Remember, relativistic corrections are needed corrections are needed because because no material no material objects can travel faster objects can travel faster than the speed of lightthan the speed of light

General Physics

Relativistic EnergyRelativistic Energy The definition of kinetic energy requires modification in The definition of kinetic energy requires modification in

relativistic mechanicsrelativistic mechanics KE = KE = mcmc22 – mc – mc22

The term mcThe term mc22 is called the is called the rest energyrest energy of the object and is independent of the object and is independent of its speedof its speed

The term The term mcmc22 depends on its speed ( depends on its speed () and its rest energy (mc) and its rest energy (mc22))

The total energy in relativistic mechanics isThe total energy in relativistic mechanics is E = KE + mcE = KE + mc22

A particle has energy by virtue of its mass alone A particle has energy by virtue of its mass alone A stationary particle with zero kinetic energy has an energy proportional A stationary particle with zero kinetic energy has an energy proportional

to its inertial massto its inertial mass

The mass of a particle may be completely convertible to The mass of a particle may be completely convertible to energy and pure energy may be converted to particles energy and pure energy may be converted to particles according to E = mcaccording to E = mc22

General Physics

Energy and Relativistic Energy and Relativistic MomentumMomentum

It is useful to have an expression relating total It is useful to have an expression relating total energy, E, to the relativistic momentum, penergy, E, to the relativistic momentum, p EE22 = p = p22cc22

+ (mc+ (mc22))22

When the particle is at rest, p = 0 and E = mcWhen the particle is at rest, p = 0 and E = mc22 Massless particles (m = 0) have E = pcMassless particles (m = 0) have E = pc

This is also used to express masses in energy unitsThis is also used to express masses in energy units Mass of an electron = 9.11 x 10Mass of an electron = 9.11 x 10-31-31 kg = 0.511 MeV/c kg = 0.511 MeV/c22

Conversion: 1 u = 931.494 MeV/cConversion: 1 u = 931.494 MeV/c22

General Physics

Mass-Energy Conservation, Pair Mass-Energy Conservation, Pair ProductionProduction

In the presence of the massive In the presence of the massive particle, an electron and a particle, an electron and a positron are produced and the positron are produced and the photon disappearsphoton disappears A positron is the antiparticle of the A positron is the antiparticle of the

electron, same mass but opposite electron, same mass but opposite chargecharge

Energy, momentum, and charge Energy, momentum, and charge must be conserved during the must be conserved during the processprocess

The minimum energy required is The minimum energy required is 2m2meecc22= 1.02 MeV= 1.02 MeV

General Physics

Mass-Energy Conservation, Pair Mass-Energy Conservation, Pair AnnihilationAnnihilation

In pair annihilation, an In pair annihilation, an electron-positron pair electron-positron pair produces two photonsproduces two photons The inverse of pair The inverse of pair

productionproduction

It is impossible to create a It is impossible to create a single photonsingle photon Momentum must be Momentum must be

conservedconserved

Energy, momentum, and Energy, momentum, and charge must be conserved charge must be conserved during the processduring the process

General Physics

Mass – Inertial vs. GravitationalMass – Inertial vs. Gravitational

Mass has a gravitational attraction for other Mass has a gravitational attraction for other massesmasses FFgg = m = mgg GM/r GM/r22

Mass has an inertial property that resists Mass has an inertial property that resists accelerationacceleration FFii = m = mii a a

The value of G was chosen to make the values The value of G was chosen to make the values of mof mgg and m and mii equal equal

General Physics

Einstein’s Reasoning Einstein’s Reasoning Concerning MassConcerning Mass

That mThat mgg and m and mii were directly proportional were directly proportional was evidence for a basic connection was evidence for a basic connection between thembetween them

No mechanical experiment could No mechanical experiment could distinguish between the twodistinguish between the two

He extended the idea to no experiment of He extended the idea to no experiment of any type could distinguish the two massesany type could distinguish the two masses

General Physics

Postulates of General RelativityPostulates of General Relativity

All laws of nature must have the same form for All laws of nature must have the same form for observers in any frame of reference, whether observers in any frame of reference, whether accelerated or notaccelerated or not

In the vicinity of any given point, a gravitational field is In the vicinity of any given point, a gravitational field is equivalent to an accelerated frame of reference without a equivalent to an accelerated frame of reference without a gravitational fieldgravitational field This is the This is the principle of equivalenceprinciple of equivalence

General Physics

Implications of General Implications of General RelativityRelativity

Gravitational mass and inertial mass are not just Gravitational mass and inertial mass are not just proportional, but completely equivalentproportional, but completely equivalent

A clock in the presence of gravity runs more slowly than A clock in the presence of gravity runs more slowly than one where gravity is negligibleone where gravity is negligible This is observed utilizing two atomic clocks, one in Greenwich, This is observed utilizing two atomic clocks, one in Greenwich,

England at sea level and the other in Boulder, Colorado at 5000 England at sea level and the other in Boulder, Colorado at 5000 feet, which confirm the predication that time slows as one feet, which confirm the predication that time slows as one descends in a gravity fielddescends in a gravity field

The frequencies of radiation emitted by atoms in a strong The frequencies of radiation emitted by atoms in a strong gravitational field are shifted to lower frequencies (shifted gravitational field are shifted to lower frequencies (shifted toward longer wavelengths)toward longer wavelengths) This has been detected in the spectral lines emitted by atoms in This has been detected in the spectral lines emitted by atoms in

massive starsmassive stars

General Physics

More Implications of General More Implications of General RelativityRelativity

A gravitational field may be “transformed away” A gravitational field may be “transformed away” at any point if we choose an appropriate at any point if we choose an appropriate accelerated frame of reference – a freely falling accelerated frame of reference – a freely falling frameframe

Einstein specified a certain quantity, the Einstein specified a certain quantity, the curvature of spacetimecurvature of spacetime, that describes the , that describes the gravitational effect at every pointgravitational effect at every point

General Physics

Curvature of SpacetimeCurvature of Spacetime

There is no such thing as a gravitational There is no such thing as a gravitational forceforce According to EinsteinAccording to Einstein

Instead, the presence of a mass causes a Instead, the presence of a mass causes a curvature of spacetime in the vicinity of the curvature of spacetime in the vicinity of the massmass This curvature dictates the path that all freely This curvature dictates the path that all freely

moving objects must followmoving objects must follow

General Physics

General Relativity SummaryGeneral Relativity Summary

Mass one tells spacetime how to curve; Mass one tells spacetime how to curve; curved spacetime tells mass two how to curved spacetime tells mass two how to movemove John Wheeler’s summary, 1979John Wheeler’s summary, 1979

The equation of general relativity is The equation of general relativity is roughly a proportion:roughly a proportion:

Average curvature of spacetime Average curvature of spacetime energy density energy density

General Physics

Testing General RelativityTesting General Relativity

General Relativity predicts that a light ray passing General Relativity predicts that a light ray passing near the Sun should be deflected by the curved near the Sun should be deflected by the curved spacetime created by the Sun’s massspacetime created by the Sun’s mass

The prediction was confirmed by astronomers during The prediction was confirmed by astronomers during a total solar eclipsea total solar eclipse

General Physics

Other Verifications of General Other Verifications of General RelativityRelativity

Explanation of Mercury’s orbitExplanation of Mercury’s orbit Explained the discrepancy between Explained the discrepancy between

observation and Newton’s theoryobservation and Newton’s theory The difference was about 43 arc seconds per The difference was about 43 arc seconds per

centurycentury

General Physics

Black HolesBlack Holes

If the concentration of mass becomes If the concentration of mass becomes great enough, a black hole is believed to great enough, a black hole is believed to be formedbe formed

In a black hole, the curvature of space-In a black hole, the curvature of space-time is so great that, within a certain time is so great that, within a certain distance from its center, all light and distance from its center, all light and matter become trappedmatter become trapped

General Physics

Black Holes, contBlack Holes, cont

The radius is called the The radius is called the SchwarzschildSchwarzschild radiusradius Also called the Also called the event horizonevent horizon It would be about 3 km for a star the size of our SunIt would be about 3 km for a star the size of our Sun

At the center of the black hole is a At the center of the black hole is a singularitysingularity It is a point of infinite density and curvature where It is a point of infinite density and curvature where

spacetime comes to an endspacetime comes to an end