Introduction to special relativity

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Index

1. Galilean-Newtonian relativity

2. Michelson-Morley experiment

3. Postulates of the special theory of relativity

4. Simultaneity

5. Time dilation

6. Length contraction

7. Velocity composition

8. Relativistic momentum

9. Equivalence between mass and energy

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1. Galilean-Newtonianrelativity

Galilean relativity principleThe basic laws of mechanics are the same in allinertial reference frames.

Inertial frameAn inertial frame is one in which Newton's first lawis valid. All inertial reference frames move withconstant velocity with respect to one another.

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(a) (b)

A person throws a ball from a moving car with velocity vball

In the reference frame of the car, the ball has velocity vball

In a reference frame fixed on the Earth, an observer measures the velocity of the ball to be vtot = vcar + vball.

Galilean-Newtonian relativity

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From the Maxwell equations, the speed of light is constant and given by

where ε0 and µ0 have the same values in all inertial reference frames.

The invariance of c contradicts classic mechanics andGalilean relativity. Physicists thought that the speedof light was equal to c only in a particular propagationmedium, which they called aether.

c 1 00

Galilean-Newtonian relativity

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According to classical mechanics the velocity of light seen by an observer should be vtot = c + vcar.

This would mean that Maxwell’s equation were no longer valid.

However, both the person on the truck and theobserver on the ground measure the speed of thelight as c, regardless of the speed of the truck.

Galilean-Newtonian relativity

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2. Michelson-Morley experiment

In order to resolve the apparent contradictionbetween Galilean relativity and Maxwell’sequations, Michelson and Morley designed anexperiment to measure the speed of the Earthwith respect to the aether (1883-1887).

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Michelson-Morley experimentA beam of light is split by mirror A into twobeams, which travel in perpendicular directions,and are reflected back to mirror A.

If the Earth is moving with velocity v relative toan aether, any difference in the times taken T1 andT2 for the beams to return to B would result in aphase difference and interference fringes in therecombined beam.

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Michelson-Morley experimentIf there is no relative motion of the Earth to anaether: v = 0 and T1 = T2 and hence the two beamswould arrive back at A in phase and reinforce eachother. No difference in phase between the twobeams was observed,

hence no relative motion and no aether

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Einstein’s theory proposed that space is not threedimensional and Euclidean, but has the fourthdimension of time, and all these dimensions under gotransformations between relatively moving referenceframes.

A full explanation of the “null” result from theMichelson-Morley experiment was provided by AlbertEinstein 18 years later:the aether could be dispensed with as no physicalexperiment can detect the absolute motion of aninertial reference frame.

3. Postulates of thespecial theory of relativity

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Postulates of thespecial theory of relativity

Principles of special relativity

1. The laws of physics are the same in all inertialreference frames.

2. Light propagates through empty space withspeed c regardless of the relative speed betweenobservers or between observer and the source.

In summary, the principles of special relativity arevalid for mechanics and electromagnetism, andthe speed of light is the same in all referenceframes.

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4. Simultaneity

As time is not absolute, one of the implications ofspecial relativity is that: observers, situated in distantplaces, do not necessarily agree on time intervalsbetween events, or on whether they aresimultaneous or not.

The simultaneity of two events depends on the reference frames in

which they are observed.

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5. Time dilation

The time it takes for light, travelling with the samespeed, to travel across a moving spaceship withvelocity v is longer for the observer on Earth than forthe observer on the spaceship. There is an experienceof time dilation.

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Time dilation

The thought experiment, using a clock consisting of alight beam and mirrors, shows that moving observersmust disagree on the passage of time.

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The time interval in the moving frame (Δt) is relatedto the time interval in the rest frame (Δt0, calledproper time) by:

The Lorentz’s factor is

and we can write

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1v2 c2

Δ t Δ t0

Time dilation

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We found that time intervals are different indifferent reference frames.

As a consequence, lengths must also be different.Length contraction is given by:

𝐋 =𝐋𝐨

𝛄

Length contraction occurs only along the directionof motion.

Analogous with the proper time, the length of anobject in its rest frame is defined as its properlength, which is equal to its length in its rest frame.

6. Length contraction16

As length shrinks and time expands, velocity alsocannot have the same composition law as in theclassical mechanics, the Lorentz transformation isused:

𝐕𝐱 =𝐕𝐱′ + 𝐕

𝟏 +𝐕𝐱′𝐕𝐜𝟐

7. Velocity composition17

The theory of special relativity means thatmomentum is also relativistic with magnitudedefined as:

𝐩 = 𝐦𝐯𝛄

Since the factor γ is always less than 1 andoccurs in the denominator, the relativisticmomentum is always larger than the non-relativistic (proper) momentum.

8. Relativistic momentum18

One of the most astonishing results of specialrelativity is that mass and energy are equivalent,in the sense that a gain or loss of mass can beequally regarded as a gain or loss of energy

If an object of mass m travelling at a speed v,Einstein showed that the total energy E of themoving object is related to its mass and speed.

𝐄 = 𝐦𝐜𝟐𝛄

9. Equivalence between

mass and energy19

When the object is at rest (v = 0 m/s), the totalenergy is called the rest energy E0:

𝐄 = 𝐦𝟎𝐜𝟐

The relationship between total energy andmomentum is:

𝐄𝟐 = 𝐩𝟐𝐜𝟐 + 𝐦𝟐𝐜𝟒

One of the important consequences of the theoryof special relativity is that objects with masscannot reach the speed c of light in a vacuum.Thus, c represents the limit of speed.

Equivalence between

mass and energy20