Relativistic velocities

According to Galilean transformation, if a ball is thrown in a moving train, an observer on the ground would measure its speed as

$$v_{\rm GROUND} = v_{\rm TRAIN} + v_{\rm BALL}.$$

What if

$$v_{\rm TRAIN} = 0.5 c{\rm\ \ \ and\ \ \ }v_{\rm BALL} = 0.6c\ ?$$

Then, according to Galileo,

$$v_{\rm GROUND} = 0.5c + 0.6c = 1.1 c.$$

But according to Einstein, this is impossible! With the Lorentz transformation

$$v_{\rm GROUND} = 0.846 c,$$

always less than the speed of light.

The Lorentz transformation has a special formula for adding relativistic speeds. If you add two speeds that are less than c, you always get a speed that is less than c.

Doppler effect

Electromagnetic radiation (light) of different kinds can be characterized by its frequency, or a wavelength. Those two are not unrelated. If you know the frequency (call it $\nu$), you can find out the wavelength (call it $\lambda$), because

$$\nu \times \lambda = c.$$

The Doppler effect is the change of the light frequency (wavelength) if the light is emitted by a source moving with respect to an observer.

It is customary to measure the Doppler effect by a quantity called the redshift, denoted by a letter z:

$$z = {\lambda_{\rm RECEIVED}-\lambda_{\rm EMITTED}\over \lambda_{\rm EMITTED}}$$

For speeds well below the speed of light,

$$z = {v\over c}.$$

For relativistic speeds the formula is more complicated. The redshift z becomes infinite when v gets close to c.

Why do scientists need the Doppler effect? It is a scientist's radar gun. With this effect, scientists can measure speeds of very distant objects in the universe.

(Of course, the policeman's radar gun also uses the Doppler effect to measure the speed of a passing car.)

Rest energy

One of the main predictions of the Einstein's SR is the rest energy.

In Newtonian mechanics, an object can have three different kinds of energy:

kinetic:

energy of motion; this energy can be extracted if an object is stopped.

internal:

energy stored inside the object as its thermal or chemical energy; this energy can only be extracted if an object is changed (say, burned down).

gravitational:

energy possed by an object acted upon by a gravity force; this energy can be extracted if an object is allowed to fall freely under the influence of the gravity.

In SR, there is a fourth kind of energy, the rest energy. Every object and body in the universe possesses an amount of energy proportional to its mass:

E₀ = mc².

WARNING: in your book the subscript 0 is at the wrong place!!!

In Newton's theory, there is no rest energy, or, more precisely, the theory does not tell anything about it.

In particle-antiparticle annihilation experiments, the SR was wonderfully confirmed!

The rest energy is enormous compared to other types of energy. The rest energy of an average person, if fully used, is enough to satisfy the energy needs of the whole world for 45 days! ¹

Another important conclusion of the Einstein's theory, is that the sum of the rest and kinetic energy of an object is simply related to the rest energy and the boost factor $\Gamma$:

$$E = \Gamma E_0 = \Gamma mc^2.$$

Thus, in order to accelerate an object to, say, 0.9c( $\Gamma=2.3$), we need to spend 130% of its rest energy!

Thus, in order to accelerate one average person to 90% of the speed of light we need to spend all the energy produced on Earth in two months!

Nuclear reactions have about 1% efficiency, i.e. mere 1% of the mass of a nuclear bomb (or nuclear reactor fuel) is converted into energy. But this is still enough to power the Sun for 10 billion years!

WARNING: the mass is the mass, it does not change when an objects moves!