# Fundamentals of Special theory of relativity

## Basics of the special theory of relativity

**Postulates of the special theory of relativity**

**The principle of relativity:**in any inertial reference system, all physical phenomena under the same initial conditions are the same, i.e. no experiments conducted in a closed system of bodies, can not be detected whether the body is resting or moving uniformly and straightforward.**The principle of the constancy of the speed of light:**in all inertial reference systems the speed of light in a vacuum is the same and does not depend on the speed of a moving light source.

Equal to the postulates of SRT, the position of the SRT on the limiting nature of the speed of light in vacuum matters: the speed of any signal in nature cannot exceed the speed of light in vacuum: *c* = 3 ∙ 10 ^{8} m / s. When moving objects with a speed comparable to the speed of light, there are various effects described below.

**1. Relativistic length reduction.**

The length of the body in the frame of reference, where it rests, is called the proper length *L *_{0} . Then the length of the body moving with speed *V* in the inertial reference frame decreases in the direction of motion to the length:

where: *c* is the speed of light in a vacuum, *L *_{0} is the length of the body in the fixed reference frame (the length of the body at rest), *L* is the length of the body in the reference frame moving with velocity *V* (the length of the body moving with velocity *V* ). Thus, body length is relative. The reduction of bodies is noticeable only at speeds comparable to the speed of light.

**2. Relativistic lengthening of the event time.**

The duration of a phenomenon occurring at a certain point in space will be the smallest in the inertial reference system relative to which this point is fixed. This means that clocks moving relative to the inertial frame of reference go slower than a fixed clock and show a longer time interval between events. Relativistic time dilation becomes noticeable only at speeds comparable to the speed of light, and is expressed by the formula:

The time *τ *_{0} , measured by the clock resting relative to the body, is called the event’s own time.

**3. Relativistic law of velocity addition.**

The law of addition of speeds in Newtonian mechanics contradicts the postulates of special relativity and is replaced by the new relativistic law of addition of speeds. If two bodies move towards each other, then their speed of convergence is expressed by the formula:

where: *V *_{1} and *V *_{2} – the speed of movement of bodies relative to the fixed frame of reference. If the bodies move in the same direction, then their relative speed:

**4. Relativistic mass increase.**

The mass of the moving body *m is* greater than the rest mass of the body *m *_{0} :

**5. The relationship of energy and body mass.**

From the point of view of the theory of relativity, body mass and body energy are practically the same thing. Thus, only the fact of the existence of the body means that the body has energy. The body has the lowest energy *E *_{0} in the inertial reference system with respect to which it rests and is called the **body’s own energy (the rest energy of the body)** :

Any change in body energy means a change in body weight and vice versa:

where: ∆ *E* is the change in the energy of the body, ∆ *m* is the corresponding change in mass. Total body energy:

where: *m* – body weight. The total energy of the body *E is* proportional to the **relativistic mass** and depends on the speed of the moving body, in this sense the following relations are important:

By the way, the kinetic energy of a body moving at a relativistic speed can be considered only by the formula:

From the point of view of the theory of relativity, the law of conservation of rest masses is unjust. For example, the rest mass of an atomic nucleus is less than the sum of the rest masses of particles entering the nucleus. However, the rest mass of a particle capable of spontaneous decay is greater than the sum of the eigen masses constituting it.

This does not mean violation of the law of conservation of mass. In the theory of relativity, the law of conservation of relativistic mass is valid, since in an isolated system of bodies, full energy is preserved, and hence relativistic mass, which follows from Einstein’s formula, thus we can speak of a single law of conservation of mass and energy. This does not mean the possibility of mass transfer to energy and vice versa.

There is a relationship between the total energy of the body, rest energy and impulse:

Photon and its properties

**Light** is a stream of quanta of electromagnetic radiation called photons. **A photon** is a particle that carries light energy. It cannot be at rest, but always moves at a speed equal to the speed of light. Photon has the following characteristics:

**1. The photon energy is equal to:**

where: *h* = 6.63 ∙ 10 ^{–34} J ∙ s = 4.14 10 ^{–15} eV ∙ s is Planck’s constant, *ν* is the frequency of light, *λ* is the wavelength of light, *c* is the speed of light in vacuum. The photon energy in Joules is very small, so for mathematical convenience it is often measured in an off-system unit – electron volts:

1 eV = 1.6 ∙ 10 ^{–19} J.

**2. The** photon moves in vacuum with the speed of light *c* .

**3. The** photon has an impulse:

**4.** A photon does not have mass in its usual sense (the mass that can be measured on scales, calculated according to the second Newton law, and so on), but according to Einstein’s theory of relativity, it has mass as a measure of energy ( *E* = *mc *^{2} ) . Indeed, anybody that has some energy has a mass. If we consider that a photon has energy, it also has a mass, which can be found as:

**5. The** photon has no electric charge.

Light has a dual nature. When light propagates, its wave properties are manifested (interference, diffraction, polarization), and when interacting with matter, particle properties (photo effect). This dual nature of light is called **wave-wave duality** .

### What is the External photoelectric effect?

**The photoelectric effect** is a phenomenon consisting of the appearance of a photocurrent in a vacuum cylinder when the cathode is illuminated with monochromatic light of a certain wavelength *λ*.

When the voltage at the anode is negative, the electric field between the cathode and the anode inhibits electrons. By measuring this **delay voltage** at which the photocurrent disappears, you can determine the maximum kinetic energy of the photoelectrons pulled out of the cathode:

Numerous experimenters established the following **basic laws of the photoelectric effect** :

- Photo effect is instantaneous. This means that electrons begin to fly out of the metal immediately after the start of irradiation with light.
- The maximum kinetic energy of photoelectrons increases linearly with the increasing frequency of light
*ν*and does not depend on its intensity. - For each substance, there is a so-called
**red border of the photoelectric effect**, that is, the lowest frequency*ν*_{min}(or the largest wavelength*λ*_{max}) at which an external photoelectric effect is still possible. - The number of photoelectrons ejected from the cathode by light in 1 s is directly proportional to the intensity of the light.

When interacting with matter, the photon completely transfers all its energy *E* = *hν to* one electron. Part of this energy an electron can dissipate in collisions with atoms of matter. In addition, part of the electron energy is expended to overcome the potential barrier at the metal–vacuum interface. To do this, the electron must perform **work exit ***A *_{out} , depending on the properties of the cathode material. The highest kinetic energy that a photoelectron emitted from the cathode may have, in this case, is determined by the law of energy conservation:

This formula is called **the Einstein equation for the external photoelectric effect** . Using the Einstein equation, one can explain all the laws of the external photoelectric effect. For the **red border of the photoelectric effect** , according to Einstein’s formula, you can get the expression:

### Bohr postulates

**The first Bohr postulate (the postulate of stationary states): the** atomic system can only be in special stationery or quantum states, each of which corresponds to a certain number *n* and energy *E _{n}* . In steady states, the atom does not radiate and does not absorb energy.

The state with the lowest energy is assigned the number “1”. It is called the **main**. All other states are assigned consecutive numbers “2”, “3”, and so on. They are called **excited**. In the ground state, the atom can be infinitely long. In the excited state, the atom lives for some time (about 10 ns) and enters the ground state.

According to the first postulate of Bohr, the atom is characterized by a system of energy levels, each of which corresponds to a particular stationary state. The mechanical energy of an electron moving in a closed path around a positively charged nucleus is negative. Therefore, the energies *E _{n}* <0 correspond to all stationary states . When

*E*≥ 0, the electron moves away from the nucleus (ionization occurs). Magnitude |

_{n}*E*

_{1}| called

**ionization energy**. The state with energy

*E*

_{1}is called the ground state of the atom.

**The second Bohr postulate (frequency rule):** when an atom transitions from one stationary state with energy *E _{n}* to another stationary state with energy

*E*, a quantum is emitted or absorbed, whose energy is equal to the energy difference of the stationary states

_{m}#### Hydrogen atom

The simplest atom is a hydrogen atom. It contains a single electron. The nucleus of an atom is a proton – a positively charged particle, whose charge is equal in magnitude to the electron charge. Usually, the electron is on the first (mostly unexcited) energy level (the electron, like any other system, tends to a state with a minimum of energy). In this state, its energy is *E *_{1} = –13.6 eV. The following ratios are fulfilled in the hydrogen atom connecting the radius of the trajectory of the electron rotating around the nucleus, its speed, and energy in the first orbit with similar characteristics in the remaining orbits:

In any orbit in a hydrogen atom, the kinetic ( *K* ) and potential ( *P* ) electron energies are associated with the total energy ( *E* ) by the following formulas:

### Atomic nucleus

It is now firmly established that the atomic nuclei of various elements consist of two particles — protons and neutrons, which are commonly called nucleons. To characterize atomic nuclei, a number of notations is introduced. The number of protons that make up the atomic nucleus is denoted by the symbol Z and is called the charge number or atomic number (this is the sequence number in the periodic table). The number of neutrons is denoted by N. The total number of nucleons (that is, protons and neutrons) is called the mass number A, for which the following formula can be written:

### Binding energy and Mass defect

The most important role in nuclear physics is played by the concept of nuclear **binding energy**. The binding energy of the nucleus is equal to the minimum energy that must be expended to completely split the nucleus into individual particles. From the law of conservation of energy, it follows that the binding energy is equal to the energy released during the formation of a nucleus from individual particles.

The binding energy of any nucleus can be determined by accurately measuring its mass. Such measurements show that the mass of any nucleus *M *_{i is} always less than the sum of the masses of the protons and neutrons it contains: *M *_{i} <Z *m *_{p}+ N *m *_{n} . The difference of these masses is called the mass **defect**, and is calculated by the formula:

The mass defect can be determined using the Einstein formula *E* = *mc *^{2} energy released during the formation of this nucleus, that is, the binding energy of the nucleus *E *_{St} :

But it is more convenient to calculate the binding energy using a different formula (here the masses are taken in atomic units, and the binding energy is obtained in MeV):

### Radioactivity. Law of radioactive decay

Almost 90% of the known atomic nuclei are unstable. The unstable nucleus spontaneously turns into other nuclei with the emission of particles. This property of the nucleus is called **radioactivity** .

**Alpha decay. **Alpha decay is the spontaneous transformation of an atomic nucleus with the number of protons Z and neutrons N into another (daughter) nucleus containing the number of protons Z – 2 and neutrons N – 2. At the same time, an *α-* particle is emitted – the nucleus of the helium atom ^{4 }_{2} He. General scheme of alpha decay:

**Beta decay. **During beta decay, an electron ( ^{0 }_{–1} e) flies out of the nucleus . Beta decay scheme:

**Gamma decay. **In contrast to *α-* and *β-* radioactivity, the *γ-* radioactivity of nuclei is not associated with a change in the internal structure of the nucleus and is not accompanied by a change in charge or mass numbers. With both *α-* and *β-*decay, the daughter nucleus can be in a certain excited state and have an excess of energy. The transition of the nucleus from the excited state to the ground state is accompanied by the emission of one or several *γ-* quanta, whose energy can reach several MeV.

**The law of radioactive decay. **Any sample of radioactive substance contains a huge number of radioactive atoms. Since radioactive decay has a random character and does not depend on external conditions, the law of decreasing the number *N* ( *t* ) of **non-decaying** nuclei at a given time *t* can serve as an important statistical characteristic of the process of radioactive decay. The law of radioactive decay has the form:

The value *T* is called the **half-life** , *N *_{0} is the initial number of radioactive nuclei at *t* = 0. The half-life is the main value characterizing the rate of radioactive decay. The shorter the half-life, the more intense the decay.

With *α* – and *β-* radioactive decay, the daughter nucleus may also be unstable. Therefore, a series of consecutive radioactive decays that result in the formation of stable nuclei are possible.

### Nuclear reactions

**Nuclear reaction** is the process of interaction of an atomic nucleus with another nucleus or an elementary particle, accompanied by a change in the composition and structure of the nucleus and the release of secondary particles or *γ-*quanta. As a result of nuclear reactions, new radioactive isotopes can form that are not found on Earth in natural conditions.

In nuclear reactions, several conservation laws are satisfied: momentum, energy, angular momentum, charge. In addition to these classical laws of conservation in nuclear reactions, the **law of conservation of the so-called baryon charge**(that is, the number of nucleons — protons and neutrons) is satisfied . For example, in a general reaction:

The following conditions are met (the total number of nucleons before and after the reaction remains unchanged):

#### Energy yield of a nuclear reaction

Nuclear reactions are accompanied by energy transformations. The energy yield of a nuclear reaction is the value:

where: *M *_{A} and *M *_{B} are the masses of the starting products, *M *_{C} and *M *_{D} are the masses of the final reaction products. The quantity Δ *M* is called the **mass defect** . Nuclear reactions can proceed with release ( *Q* > 0) or with energy absorption ( *Q* <0). In the second case, the initial kinetic energy of the initial products should exceed the value of | *Q* |, which is called the **reaction threshold** .

In order for a nuclear reaction to have a positive energy yield, the specific binding energy of the nucleons in the nuclei of the initial products must be less than the specific binding energy of the nucleons in the nuclei of the final products. This means that Δ *M* must be positive.

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