The Theory of Special Relativity. Learning Objectives Einstein’s two postulates in his theory of special relativity: The principle of relativity. (Same

The Theory of Special Relativity

Learning Objectives Einstein’s two postulates in his theory of special relativity:

The principle of relativity. (Same principle as in Newtonian physics)The constancy of the speed of light. (Breaks from Newtonian physics)

Using Einstein’s two postulates, derive space and time transformations between inertial reference frames (derived transformations are same as the Lorentz transformations):




Einstein’s Postulates The theory of special relativity was introduced in Einstein’s paper “On the

Electrodynamics of Moving Bodies” in 1905. In this paper, Einstein made two postulates:

1. The Principle of Relativity. The laws of physics are the same for all observers in uniform motion relative to one another (i.e., in all inertial reference frames).

2. The Constancy of the Speed of Light. Light moves through a vacuum at a constant speed c that is independent of the motion of the light source relative to the observer.

The first principle is also postulated in Newtonian physics (as expressed in the Galilean transformations). In Newtonian physics, however, the second postulate contradicts the first, and therefore marks a break from Newtonian (classical) physics.




To see how Einstein derived the Lorentz transformations from his two postulates, let us write down the most general set of transformation equations between the space and time coordinates (x, y, z, t) and (x´, y´, z´, t´) of the same event measured from two inertial reference frames S and S´ in uniform relative motion

Space and Time Transformations

Say that a light bulb at rest in the S frame, located at (x, y, z), flashes at a time (t). As measured in the S´ frame, what is the corresponding location (x´, y´, z´) and time (t´) when the light bulb flashes?

In Newtonian physics, what would be the values of the coefficients?

Notice that the transformation equations are linear. Why would non-linear transformations not make sense?


Say that a light bulb at rest in the S frame, located at (x, y, z), flashes at a time (t). As measured in the S´ frame, what is the corresponding location (x´, y´, z´) and time (t´) when the light bulb flashes?

If the transformations were non-linear, the displacement between two objects or the time interval between two events would then depend on the choice of origin for the S (and hence S´) frame. This is unacceptable as the laws of physics cannot depend on the numerical coordinates of an arbitrarily chosen coordinate system.


Einstein’s first postulate, the Principle of Relativity, implies that lengths perpendicular to u are unchanged; i.e., , y = y´ and z = z´.


To see this, imagine that frame S has a meter stick oriented along the y-axis with one end at the origin. Imagine also that frame S´ has a meter stick oriented along the y´-axis with one end at the origin.

Imagine further that paint brushes are mounted perpendicular at both ends of each meter stick, and that frames S and S´ are separated by a sheet of glass.

If an observer in frame S sees the meter stick to be shorter (or longer) in frame S´, the line drawn by the paintbrush in frame S´ will be inside (outside) the line drawn by the paintbrush in frame S.





Similarly, if an observer in frame S´ sees the meter stick to be shorter (or longer) in frame S, the line drawn by the paintbrush in frame S will be inside (outside) the line drawn by the paintbrush in frame S´.





Both sets of lines cannot lie inside (outside) the other, so the only possible outcome is that both lines must overlap; i.e., lengths along y and y´ do not change.

The same arguments apply for lengths along z and z´.


Thus, in the general set of transformations

the Principle of Relativity implies that




the Principle of Relativity implies that


Measurement of time in Eq. (4.9) must give the same result if y is replaced by -y or z is replaced by -z; i.e., time measurement cannot depend on the side of the x-axis on which the event occurs.


the above argument implies that


Measurement of time in Eq. (4.9) must give the same result if y is replaced by -y or z is replaced by -z; i.e., time measurement cannot depend on the side of the x-axis on which the event occurs.


the above argument implies that


Consider the motion of the origin O´ of the frame S´ relative to the frame S.

The clocks in both frames are synchronized at t = t´ = 0 when the origins O and O´ coincide.

After a time t has elapsed as measured in frame S, the origin O´ is located at x = ut. Of course, in frame S´, the origin O´ is always located at x´ = 0.

In this situation, Eq. (4.6)

becomes

which implies that


Consider the motion of the origin O´ of the frame S´ relative to the frame S.

The clocks in both frames are synchronized at t = t´ = 0 when the origins O and O´ coincide.

After a time t has elapsed as measured in frame S, the origin O´ is located at x = ut. Of course, in frame S´, the origin O´ is always located at x´ = 0.

In this situation, Eq. (4.6)

becomes

which implies that


Collecting the results thus far, Eqs. (4.6-4.9) have been reduced to

In Newtonian physics, what would be the values of the coefficients?



In Newtonian physics, what would be the values of the coefficients?a11 = 1, a41 = 0, and a44 = 1, which reduces the above equations to the Galilean transformation.



So far we have only applied one of Einstein’s postulate, the Principle of Relativity, which also is a basic principle of Newtonian physics.

As we shall see, when we apply Einstein’s 2nd postulate, Eqs. (4.10-4.13) do not reduce to the Galilean transformations.


Einstein’s 2nd postulate, the constancy of the speed of light, implies that all observers measure exactly the same value for the speed of light.

Suppose that a lightbulb, located at the origins O in the reference frames S, is turned on at time t = t´ = 0 when the origins O and O´ coincide.

After a time t has elapsed in the S reference frame, an observer in this frame would see a spherical wavefront with a radius ct moving away from the origin O with speed c and satisfying


Equation for sphere

In Newtonian physics, what would an observer in the S´ frame see for the same wavefront? What equation would this wavefront satisfy?





Equation for sphere

In Newtonian physics, what would an observer in the S´ frame see for the same wavefront? What equation would this wavefront satisfy?

(x´+ut)2 + y´2 + z´2 = (ct´)2




Similarly, after a time t´ has elapsed in the S´ reference frame, an observer in this frame would see a spherical wavefront with a radius ct moving away from the origin O´ with speed c and satisfying


Equation for sphere

Equation for sphere

Inserting Eqs. (4.10-4.13) into Eq. (4.15)

and comparing the result with Eq. (4.14),

we find that


Assignment question

Thus, if the speed of light is constant in all reference frames, we get

which are the same transformations as the Lorentz transformation equations. If you differentiate x´ wrt to t´ (as we will do later), you will find that the speed of light is constant in all reference frames.


Compare Lorentz with Galilean transformations:



Say that u = 0.1c. If an object is located at x = 0 in frame S, at what x´ will it be located in frame S´ after t = 1 s according to the Galilean transformation?




x´ = 0 – 0.1 c × 1 s = -0.1 c × 1 s




x´ = 0 – 0.1 c × 1 s = -0.1 c × 1 s

According to the Lorentz transformation?




x´ = 0 – 0.1 c × 1 s = -0.1 c × 1 s

According to the Lorentz transformation?x´ = (0 – 0.1 c × 1 s) / √ (1 – 0.12)

= (-0.1 c × 1 s) / 0.995 < (-0.1 c × 1 s)

This reflects the contraction of space along the x´ direction in frame S´ according to S.







x´ = 10 – 0.1 c × 1 s




x´ = 10 – 0.1 c × 1 s





x´ = 10 – 0.1 c × 1 s

According to the Lorentz transformation?x´ = (10 –0.1 c × 1 s) / √ (1 – 0.12)

= (10 – 0.1 c × 1 s) / 0.995 > (10 –0.1 c × 1 s)

This reflects the contraction of space along the x´ direction in frame S´ according to S.



Why does t´ depend on x?



Say that u = 0.1c. Suppose two flashbulbs, one located at x = 10 and the other at x = 20, both flash at time t = 10 s in frame S. At what time t´ would they flash in frame S´ according to the Galilean transformation?


Implicit in this equation is that t´ = t at x = 0; i.e., where origins of S´ and S frames coincide.


Say that u = 0.1c. Suppose two flashbulbs, one located at x = 10 and the other at x = 20, both flash at time t = 10 s in frame S. At what time t´ would they flash in frame S´ according to the Galilean transformation? Both flash at t´ = 10 s






According to the Lorentz transformation?t´1 = (10 − 0.1 c 10 / c2) / √ (1 – 0.12)

= (10 − 1 / c) / √ 0.995

t´2 = (10 − 0.1 c 20 / c2) / √ (1 – 0.12) = (10 − 2 / c) / √ 0.995

They do not flash at the same time in frame S´. This is the concept of the loss of simultaneity.





According to the Lorentz transformation?t´1 = (10 − 0.1 c 10 / c2) / √ (1 – 0.12)

= (10 − 1 / c) / √ 0.995

t´2 = (10 − 0.1 c 20 / c2) / √ (1 – 0.12) = (10 − 2 / c) / √ 0.995

The reason that t´1 ≠ t and t´2 ≠ t reflects the loss of synchronicity because the clock in frame S´ appears to tick slower according to observer in frame S.




Why does t´ depend on x? Reflects concept of the loss of simultaneity.



Under what condition does the Lorentz transformation closely approach the Galilean transformation?



Under what condition does the Lorentz transformation closely approach the Galilean transformation? When u « c


The Lorentz Transformations

Hendrik A. Lorentz, 1853-1928Half-silvered mirror

Light source

Direction of Earth’s motion through etherMatter at rest in ether Matter moving to right

with respect to ether

The Lorentz transformations were originally proposed by the Dutch physicist Hendrik Antoon Lorentz and his collaborators. They believed in the luminiferous ether hypothesis, and were motivated purely by the desire to find a mathematical transformation under which Maxwell’s equations were invariant when transformed from the ether to a moving reference frame (i.e, so that light travels at the same speed no matter the motion of the observer relative to the ether).

Hendrik A. Lorentz, 1853-1928

The Lorentz Transformations

Matter at rest in ether Matter moving to right with respect to ether

Lorentz believed that space is an absolute, and that objects contract along the direction of motion such that the speed of light is a constant no matter our speed relative to the ether.

Einstein discarded the notion that space (and time) is an absolute, but instead has dimensions that depend on our relative motion.

Hendrik A. Lorentz, 1853-1928

Albert Einstein, 1879-1955

The Lorentz transformations:

The factor

is called the Lorentz factor, and is often used to estimate the importance of relativistic effects. When u << c, γ ≈ 1 and the Lorentz transformation reduces to the Galilean transformation. When u → c, γ > 1.

γ = 1.00001 when u = 1/222 c = 1341 km/s, ~170 times the speed of the space shuttle (~28000 km/h = 7.8 km/s).

γ = 1.01 when u = 1/7 c.

γ = 1.1 when u = 5/12 c.

The γ factor

The γ factor

u/c 0.14 1.010.42 1.100.55 1.200.64 1.300.70 1.400.75 1.500.87 2.00.94 3.00.98 5.00.99 10.0

-

What if the reference frame is moving in the –x-direction relative to us? We then use the inverse Lorentz transformations, which can be derived in a similar manner as the (forward) Lorentz transformations, or using symmetry arguments by simply switching primed and unprimed quantities and replacing u with –u.

The inverse Lorentz Transformations

Assignment question

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The Theory of Special Relativity. Learning Objectives Einstein’s two postulates in his theory of special relativity: The principle of relativity. (Same