Special RelativitySpecial Relativity

Einstein’s revolution - rethink the meaning of space and time

Published “Special Theory of Relativity” in 1905 followed by “General Theory” in 1916 which incorporates gravity

He began with the postulate that the laws of physics should be independent of the velocity of the observer

Einstein’s revolution - rethink the meaning of space and time

Published “Special Theory of Relativity” in 1905 followed by “General Theory” in 1916 which incorporates gravity

He began with the postulate that the laws of physics should be independent of the velocity of the observer

FoundationsFoundations

Applying this postulate to Maxwell’s theory of EM radiation requires a solution to the equations that is constant in time (for someone moving at light speed) but sinusoidal in space - not possible!

See http://www.phys.unsw.edu.au/einsteinlight/ for a discussion of the invariance of the speed of light.

Thus, speed of light must be the same for all observers, independent of their motion (EM waves are different from mechanical waves).

How can velocity of light be constant? Velocity measurement depends on distance and time intervals - could these quantities depend on the motion of the observer?

A simple experiment shows that simultaneity is not absolute. Thus, time is not absolute.

Einstein then investigated how different types of situations appear to observers with different velocities (inertial reference frames).

Einstein’s postulate could be stated “There is no experiment we can perform to tell us which inertial frame is moving and which is at rest.” There is no ‘preferred’ inertial frame.

Absolute time absolute simultaneity

The constancy of the speed of light paradox

Imagine a red dot emits a flash of light while a blue dot is moving away from the red one at half the speed of light.

The red dot sees itself at the center of the expanding sphere of light.

SR insists that the blue dot also sees the light moving outward at the same speed in all directions.

How can that be so? Paradox!

(http://casa.colorado.edu/~ajsh/sr/paradox.html)

ChallengeFind the solution to this paradox (i.e. arrange it so that both Red and Blue regard themselves as being in the center of the sphere of light).

In a spacetime diagram, the units of space and time are chosen so that light goes one unit of distance in one unit of time (i.e. c = 1). Light moves upward and outward at 45 degrees in the spacetime diagram.

The lines along which Red and Blue move are called worldlines. Each point in 4-dimensional spacetime is called an event. Light signals converging to or expanding from an event follow a 3-dimensional hypersurface called the lightcone. In the diagram, the sphere of light expanding from the emission event is following the future lightcone. There is also a past lightcone not shown here.

Spacetime diagram of Red emitting a flash of light. Time moves vertically while space dimensions are horizontal.

Einstein's solution to the paradox is that Blue’s spacetime is skewed compared to Red's.

Notice that Blue is in the center of the lightcone, according to the way he perceives space and time.

Red remains at the center of the lightcone according to the way she perceives space and time.

From Blue's point of view, his spacetime seems normal and Red’s spacetime is skewed.

The Solution

Time DilationTime Dilation

Imagine a clock being observed from two different inertial reference frames. In one, the clock is at rest wrt the observer (proper time) and the other sees the clock moving with some velocity.

Proper time is the time interval between two events at the same place, Δt (the time viewed in the frame in which the clock is at rest). The time viewed in the frame moving wrt to the clock we will call Δt’.

Imagine a clock being observed from two different inertial reference frames. In one, the clock is at rest wrt the observer (proper time) and the other sees the clock moving with some velocity.

Proper time is the time interval between two events at the same place, Δt (the time viewed in the frame in which the clock is at rest). The time viewed in the frame moving wrt to the clock we will call Δt’.

Red and Blue have identical clocks consisting of a light beam bouncing off a mirror.

If Red and Blue remain at rest relative to each other, both agree that the clocks run at the same rate.

Now let Blue move away at velocity v relative to Red, in a direction perpendicular to the direction of the mirror.

For Blue, his clock runs at the same rate as before.

But from Red’s point of view, although the distance between Blue and his mirror at any instant remains the same as before, the light has further to go. And since the speed of light is constant, Red thinks it takes longer for Blue’s clock to run than her own.

Thus Red thinks Blue's clock runs slow relative to her own.

How much slower?

For Red, the time it takes for the lightbeam to travel to the mirror and back is Δt = 2L/c.

But, Red thinks the distance traveled by the light beam between Blue and his mirror is something slightly more than L call this unit D. The time for Blue’s lightbeam to travel to the mirror and back is Δt’ = 2D/c.

Blue is moving at speed v, so Red thinks he moves a distance of ½ v Δt’ during the time taken by the light to travel from Blue to his mirror.

Using the Pythagorean theorem we get:

L D

½ v Δt’

Substituting this D into the previous equation gives…

= 1/(1-2)1/2 where = v/c

Lorentz gamma factor

Lorentz gamma factor, introduced by the Dutch physicist Hendrik A. Lorentz in 1904, one year before Einstein proposed his theory of special relativity.

Δt’ = Δt

The time interval measured in the frame in which the clock is moving is greater than that in which the clock is at rest

And with the definition of Δt, we get

Red thinks Blue's clock runs slow. From Blue's perspective it is Red who is moving, and Red whose clock runs slow. How can both think the other's clock runs slow? Paradox!

The resolution involves simultaneity.(use this spacetime diagram to help)

While Red thinks events happen simultaneously along horizontal planes in his diagram, Blue thinks events occur simultaneously along skewed planes. Thus Red thinks her clock ticks when Blue is at the point , before Blue's clock ticks. Conversely, Blue thinks his clock ticks when Red is at the point , before Red's clock ticks.

Question: How fast must a particle travel to live 10 times as long as the same particle at rest?

Δt’ = Δt

Length ContractionLength Contraction

Length is determined by measuring the positions of two ends and taking the difference

Measurements must be carried out simultaneously

But, observers in different inertial frames cannot agree on simultaneity of events separated in space!

Thus, lengths appear different in different inertial frames

Length is determined by measuring the positions of two ends and taking the difference

Measurements must be carried out simultaneously

But, observers in different inertial frames cannot agree on simultaneity of events separated in space!

Thus, lengths appear different in different inertial frames

Proper Length, L - length of object at rest let L’ denote the length observed from an inertial frame

For a stick moving with velocity v, the time interval (measured in the non-moving frame) between measurement of the front and the back at a single marker as it passes by is

t = L’/vIn the frame of the moving stick (riding along with it)

L = vt’time dilation relates the two time intervals t’ = t

Rearranging, we get L’ = L/ - Lorentz contraction

Length contraction is symmetric - a person in either reference frame will observe lengths contracted in the other frame.

Note - only lengths in direction of motion are contracted

Length contraction leads to another paradox!

Thought experiment - the Ladder Paradox. If a ladder travels horizontally it will undergo a length contraction and will therefore fit into a garage that is shorter than the ladder's length at rest. On the other hand, from the point of view of an observer moving with the ladder, it is the garage that is moving and the garage will be contracted. The garage will therefore need to be larger than the length at rest of the ladder in order to contain it.

How is this so since if the ladder fits into the garage in one reference frame, it must do so in all?

The solution:What one observer (e.g. the garage) considers as simultaneous does not correspond to what the other observer (e.g. the ladder) considers as simultaneous events.

Doppler ShiftDoppler Shift

Important for astronomy since most astrophysical objects are studied using emitted light and the motion of source is determined by the Doppler shift of the wavelength of light

Doppler shifts experience relativistic effects since wavelengths involve length and frequencies involve time

Does not matter whether the source or the observer is the one moving

First, a brief reminder – what is the Doppler shift?

Important for astronomy since most astrophysical objects are studied using emitted light and the motion of source is determined by the Doppler shift of the wavelength of light

Doppler shifts experience relativistic effects since wavelengths involve length and frequencies involve time

Does not matter whether the source or the observer is the one moving

First, a brief reminder – what is the Doppler shift?

Wave Characteristics:

•Wavelength - Distance between successive wave peaks •Period – Time between passing wave peaks•Frequency – Number of wave peaks passing per unit time (1/Period)•Wave Speed – wavelength x frequency

Visible light Visible light is a is a smallsmall part of the part of the electro -electro -magnetic magnetic spectrum.spectrum.

Doppler Effect - Doppler Effect - SoundSound

The Doppler Shift: LightThe Doppler Shift: Light

Stationary source:

Moving source:

How much shorter or longer the observed wavelength is depends on the speed of the emitting source.

obs - emit

emit=

v

c

For sources moving away from us

or + 1obs

emit

=v

c

Moving Light Source (can get the same thing with moving observer!)

•Assume source is moving away from the receiver (us) at speed v•Source emits N waves in time t (measured by receiver)

•First wave travels ct and source travels vt•Wavelength is the distance between the source and the first wave divided by the number of waves

o = (ct + vt) / N = (t/N)(c+v)

•Relate this wavelength to that in the emitting source reference frame

e = (cte) / N

Time dilation relates the two time intervals te = t /

e = (c/N)(t/) •Use this to eliminate t in first eq. above to get

o/e = [1 + (v/c)] (like classical Doppler but with extra factor)

If we consider the source and observer moving towards each other o/e = [1 - (v/c)]

So, in general o/e = [1 ± (vR/c)]

The “v” here is actually “vR” in the radial direction. The Lorentz factor γ also contains a “v” which is the total velocity which effects time dilation.

Problem: Find the wavelength at which we will observe the Hα emission line (λ = 656 nm) if it is emitted by a galaxy moving away with v/c=0.3?

For objects moving away with speeds close to c (where radial

velocity is close to total velocity) we can simplify this equation to

o/e = (1+ )1/2

where = v/c(1 - )1/2

Comparison of Classical vs. Relativistic Doppler Shift

Redshift (z) is often used in astronomy to measure the ratio of o/e

z = o/e – 1

For sources moving at non-relativistic speeds z ~ v/c but quickly deviates from this when v is greater than 20% c.

Space TimeSpace Time

These phenomena are not illusions but real effects

Unlike classical physics, Einstein realized that space and time were intertwined with the laws of physics, not just an absolute grid on which the laws were laid

It helps to stop thinking in terms of 3-d space alone and adding the 4th dimension of time. Time is just treated as an additional dimension much like space.

These phenomena are not illusions but real effects

Unlike classical physics, Einstein realized that space and time were intertwined with the laws of physics, not just an absolute grid on which the laws were laid

It helps to stop thinking in terms of 3-d space alone and adding the 4th dimension of time. Time is just treated as an additional dimension much like space.

Four Vectors and Lorentz Transformations

•Let time be denoted by ct so that it has the same units as the other spatial dimensions•In space-time, use four-vectors to denote an event (ct,x,y,z)•Observers in different inertial frames will note different coordinates for events, but the coordinates are related.•Example: one frame at rest and one moving with velocity v in direction x

•These relationships are called the Lorentz Transformation

•In special relativity, the Lorentz Transformation is just the transformation between the spacetime frames of two inertial observers.

•In general, a Lorentz Transformation consists of a spatial rotation about some spatial axis, combined with a Lorentz boost by some velocity in some direction

•Only space along the direction of motion gets skewed with time. Distances perpendicular to the direction of motion remain unchanged.

The effect of the Lorentz transformation is to rotate the axes (ct and x) through an angle whose tangent is v/c. The unusual feature is that the two axes rotate in opposite directions so that they are no longer perpendicular.

Invariance under rotation

Length intervals are invariant under the rotation

of a normal spatial grid.

Since the LT has properties of rotation, we find that the spacetime interval is invariant under the transformation, just as length intervals are invariant under the rotation of a normal spatial grid

(s)2 = (ct)2 - (x)2 - (y)2 - (z)2

Length contraction and time dilation can be derived from this invariance - integral parts of the nature of spacetime!

3 Types of spacetime intervals:

(s)2 = zero - lightlike (photons travel along these lines - this is a lightcone in 2-d space)

(s)2 > zero - timelike (positions are close enough in space that a photon would have had more than enough time to travel from one event to the other)

(s)2 < zero - spacelike (photon cannot traverse the distance in the time given - one event could not have caused the other)

Energy and Momentum

•Much like the coordinates of spacetime transform according to the Lorentz transformation, so do energy and momentum•Why is this the case? Consider how the energy and momentum of a photon are related E=cpx for a photon moving in the x direction (similar to the relation between position and time x=ct)•The energy-momentum four vector (like the spacetime four vector) is (E, cpx, cpy, cpz) and the Lorentz transformations are

If E’ is energy at rest (where px’ = 0)then let E’ be Eo and

E = Eo is the relativistic energy

Now we see that Eo can’t be 0 or else the particle’s energy would always be zero, regardless of its velocity, and we know that’s not true. So particles must have non-zero rest energy.

From the transformations we get the relativistic momentum cpx = Eo

In the non-relativistic limit (=1), the momentum should be the classical expression

px = mvx

Putting this into the above, we get Eo = moc2 (where mo is the rest mass of the particle)

For relativistic energy and momentumE = moc2 p = mov

What happens to the energy quantity as v approaches c?E goes to infinity and thus it takes an infinite amount of energy to accelerate a particle (with a non-zero mass) to the speed of light.

Energy-momentum four-vectors also have an invariant lengthE2 - (cpx)2 - (cpy)2 - (cpz)2

Evaluated in the rest frame of some particle, momentum is zero and the energy is just moc2, which is invariant for any observer.

Particles that can travel faster than the speed of light?

Tachyons - they can never go slower than c if they exist(c is also a limiting factor for them)

If they exist, they could interact with photons and be observable, but no experiments thus far have found them...

Neutrinos – in recent years, found traveling faster than light. But results could be explained by SR (clocks measuring times in orbit moving relative to experiment), though experimental flaws were also blamed…

What is the rest energy of a proton? mp = 1.67x10-24 g