# Any particle has a velocity of zero

One of the most familiar expressions in Galilean physics is that of the **kinetic energy** of a moving body: mv^{2}/ 2. However, this cannot be maintained within the framework of the special theory of relativity: The kinetic energy can be interpreted as the work that has to be expended to accelerate an initially resting body in order to bring it to speed v. According to the traditional formula, it would be possible to bring a body to (or above) the speed of light with a finite amount of work. The speed of light as the upper limit (which we discussed in the section on time dilation) must also be reflected in a meaningful concept of energy in some way.

Therefore the question arises *whereby* the term mv^{2}/2*replaced* shall be.

A simple butting process |

In order to develop an energy concept that fits the basic structure of the special theory of relativity, we proceed according to a method that has already been tried and tested: We will consider a simple process from different points of view (i.e. in different inertial systems). This is the following butting process:

Two bodies of the same rest mass m move towards each other with velocities v and -v. In the event of an impact, they stick to each other, so that a static body of mass M is created. (That's what they call one **inelastic collision**. We can best imagine that the balls are made of wax):

Now one would say right away that M = 2m, because the resulting body is composed of two objects of mass m. According to Galilean physics, this is also the case. But we saw in the previous section that the mass is less constant than you might have thought, so we are careful and leave M open for the time being.

Let us first analyze the process in the context of the *Galilean physics*. The total momentum of the (closed) system is preserved: before and after the collision it is zero. We have to be careful when conserving energy: Before the collision, the total energy consists only of the kinetic energy of the two bodies: E._{in front} = 2 mv^{2}/ 2 = mv^{2}. After the collision, the kinetic energy of the resulting body is zero because it is at rest. On the other hand, the total energy of a closed system does not change, so we have to conclude that the total energy after the collision, E_{to}, also by mv^{2} given is. It is obviously contained in the resulting body and is called **Interaction energy**. (For example, it may have been used to deform the wax and converted into heat). So we hold on to the fact that the interaction energy occurring in this collision process goes through

W = m v^{2} | (1) |

given is.

On the way to a new energy concept |

Now let's analyze the situation *relativistic*. The sizes m and M should stand for the rest masses of the occurring objects. In the inertial system I, which corresponds to the figures, there is nothing exciting to learn: The relativistic total momentum (which, as we formulated in the previous section, is obtained) is zero before and after the collision, and we know about the relativistic energy nothing yet.

We want the same process from one now *other* Consider the inertial system I ', namely that in which the *right* of the two bodies flying onto each other *in front* the shock *in peace* is. We will then from the (*already known*) Theorem of the conservation of the relativistic momentum and the (*already known*) Conversion of the occurring speeds into the system I 'a *new knowledge* that will show us the way to the relativistic energy concept.

The system I 'moves in I with velocity -v. How fast does the left body move in it? According to Galilean physics, its speed would be given by 2v, but in the theory of relativity we have to apply the formula for the relativistic addition of speed: Then the speed we are looking for is

u = (v + v) / (1 + v^{2}/ c^{2 }) = 2v / (1 + v^{2}/ c^{2 }) , | (2) |

is therefore less than 2v. The situation before the collision looks like this in I ':

After the impact, the resulting body moves with velocity v:

Now we apply the theorem of the conservation of relativistic momentum in this inertial system. The momentum is (see formula (4) in the previous section) before the collision through m u (1 - u^{2}/ c^{2 })^{-1/2}, after the impact through M v (1 - v^{2}/ c^{2 })^{-1/2} given. The preservation of the relativistic impulse therefore demands

m u (1 - u^{2}/ c^{2 })^{-1/2}= M v (1 - v^{2}/ c^{2 })^{-1/2} . | (3) |

In this equation everything is known except for M. (If we go through the same line of argument in Galilean physics, we get a corresponding equation, which is reduced to the statement M = 2m). So we can use equation (3) to calculate M. We solve it for M and use (2) to express u by v. After a short calculation it turns out

M = 2 m (1 - v^{2}/ c^{2 })^{-1/2} . | (4) |

The mass of the resulting body is actually greater than the initially assumed value of 2m. What does that mean? If the velocity v is not too great (i.e. v / c small), we can use the approximation formula (1 - x^{ })^{-1/2} »Use 1 + x / 2, which applies to small x (these are the first two terms of the Taylor series) and obtain

M »2 m + mv^{2}/ c^{2} | (5) |

We remember (1) and realize that (5) is also called

M »2 m + W / c^{2} | (6) |

can be written. (The use of the non-relativistic formula (1) as an approximation is certainly justified if v is not too large). That means the mass of the resulting body compared to *Sum of the masses of the components* (the Galilean value) is increased by the contribution "interaction energy / square of the speed of light". With this we are on the trail of a pattern that applies very generally: **If the energy DE is supplied to a system, its mass changes** by the amount

Dm = DE / c^{2} . | (7) |

In the case of our impact process, the energy DE = W (e.g. as deformation energy) has been added to the system, which consists of the two bodies adhering to each other after the impact. The fact that in (5) there is only an »instead of an equal sign is explained by the fact that W was expressed by the non-relativistic formula for the kinetic energy. In the light of (7), however, an equal sign could be written in (6) if W is not interpreted as a non-relativistic expression adopted from (1), but as relativistic deformation energy. If you feel like it, you can calculate the latter exactly with it.

We do not try to justify the mass change (7) more precisely (although there are various arguments for it), but take it as a hypothesis that gives rise to further speculation: If **every change in energy leads to a change in mass**, so it could also be used for the relativistic mass increase through movement, which we discussed in the previous section:

Compared to the state of rest, a moving body has its kinetic energy E_{kin} has been fed. On the other hand, its mass has also changed from the rest mass m to the dynamic mass m_{dyn} (see formula (3) of the previous section) changed: Dm = m_{dyn} - m. If we combine this with (7), we get m_{dyn} - m = E_{kin}/ c^{2}i.e.

E._{kin} = (m_{dyn}- m) c^{2} = E - m c^{2} , | (8) |

in which

| (9) |

is. We define the expression (8) as that **relativistic kinetic energy** of a body moving with velocity v. If v is small compared to the speed of light, the approximation formula (1 - x^{ })^{-1/2} »1 + x / 2, can be used for small x, and (8) is reduced to the non-relativistic expression mv^{2}/ 2. For speeds close to c, E becomes arbitrarily large.

Einstein took this line of thought even further: If the two greats **energy** and **Dimensions** are generally coupled to each other according to (7), we may not really need to distinguish them! They would be to each other "**equivalent to**", not to say identical. (This postulate is called"**Equivalence of mass and energy**"). Thus the expression (9) would be a *better candidate* for a relativistic concept of energy as (8): Energy and (dynamic) mass would then be proportional to each other, i.e. up to the factor c^{2} with each other too **identify**. Mass would basically be seen as a special form of energy.

The expression (8) for the kinetic energy looks a bit strange indeed: Actually, the quantity (9) is more like many of the other formulas that we derived in this course. There are a number of other arguments (which are a bit more mathematical and which we will not go into here) in favor of the fact that E represents the total energy of a body (in the absence of forces). This size bears the simple and concise name **relativistic energy**.

If E represents the total energy of a body (in the absence of forces) then E = mc^{2} + E_{kin}, i.e. the body already has an energy due to its mass - even when it is at rest. We can also see this by inserting v = 0 in (9). A resting body of (rest) mass m then has the energy

E._{O} = m c^{2} . | (10) |

She will **Resting energy** called.

At first glance, these considerations certainly seem adventurous. However, there are numerous arguments that lead to the same consequences. For example, the expression (4), which is on a safe footing, is a subsequent justification of the definition (9): Mc^{2} would then be as energy *to* to interpret the impact, and this quantity is equal to the sum of the energies (9) for the two approaching particles, i.e. the total energy of the system *in front* the shock. This is a property that we must demand for any reasonable concept of energy: no energy must be lost in a closed system.

This brings us to the question of how the relativistic energy is in *experimentally verifiable* Way expresses.

The most important such statement that the **relativistic mechanics** about the **relativistic energy** makes is the set of theirs **conservation**:

In a system of mass points that interact with each other in any way, but which is entirely closed (i.e. does not interact with its surroundings), the **Sum of the relativistic energies of all particles** not over time. The relativistic energy of a particle means the expression given by (9), whereby all movements are related to an arbitrary inertial system.

This statement - together with the theorem of the conservation of relativistic momentum, which we discussed in the previous section - plays an important role in the analysis of elementary particle processes. It has been excellently confirmed by measurements in particle accelerators. The same experimental results show that the sum of the kinetic energies (8) in elementary particle processes *Not* is always preserved. So we now have the experimental confirmation that the quantity (9) - instead of (8) - represents the correct relativistic energy. The kinetic energy is merely *a* Part of the total energy. The *other* Part is the rest energy.

The importance of resting energy |

In contrast to non-relativistic mechanics, relativistic mechanics is the sum of all those involved in a process **Rest masses Not receive**. An example is the collision process considered above, in which the sum of the rest masses before the collision is 2m, but M after the collision, which differs from 2m according to (4). In nature there are transformations of elementary particles in which the sum of the rest masses also occurs

*Not*is preserved. If the interaction energy occurring in such a process is negative, one speaks of a

**Binding energy**which, according to (7), leads to a decrease in the sum of the rest masses and, from an energetic point of view, is responsible for the stability of particles.

If the sum of the rest masses becomes smaller in a process, the sum of the kinetic energies must increase. In the extreme case this can mean that a system of mass m can go through a process in which kinetic energies of the order of magnitude mc^{2} occur. Just calculate the rest energy of a pencil! If even a small part of this energy occurs as kinetic energy (or is converted into other forms of energy), this process will be like a huge explosion. This is the principle on which the hydrogen bomb (nuclear fusion), the atomic bomb (uncontrolled nuclear fission) and the generation of energy in nuclear power plants (controlled nuclear fission) are based.

Energy, work and c as the top speed |

The kinetic energy of a body moving with velocity v is - in non-relativistic as in relativistic mechanics - nothing else than that **job**that has to be plugged into it in order to accelerate it from the state of rest to speed v. This opens up another possibility to test the relativistic energy concept experimentally. For example, electrically charged particles are accelerated - as the name suggests - through the effect of electromagnetic fields in accelerator facilities. Energy has to be put into them. This energy makes itself felt as the work that is necessary to generate the fields. It can be measured and compared with expression (8).

There is an interesting consequence: In order to accelerate such a particle (which has a non-vanishing rest mass) to the speed of light, an infinite amount of energy would be necessary according to formula (8) (since the denominator of E would then be zero). To bring it very close to c, a lot of energy is required - more than would be expected from non-relativistic mechanics. That gives a measurable effect! If more energy is pumped into the particle, its speed approaches the speed of light more and more, but never reaches it.

Here we have - in addition to another experimental test of the special theory of relativity - the dynamic reason that **a massive particle can never reach the speed of light**(See c for maximum speed in the section on time dilation).

Now it's time to talk about the so-called "**massless**** Particle**In nature there are particles that move at the speed of light. At least photons do. (Until recently, neutrinos were one of them, but that is no longer certain). In any case, the special theory of relativity deals with such Particles some basic statements that we will now reproduce, but not substantiate any further.

There is no fundamental argument against particles moving at the speed of light. However, such particles can **no rest mass** be assigned. (One can also say that their rest mass is zero, hence the name "*massless* Particle "). You **move** (due to the constancy of the speed of light) in *each* Inertial system **at the speed of light**, therefore - in contrast to massive particles - have no rest system.

The formulas for momentum and energy derived in the previous and in this section do not apply to them. Nevertheless one can assign a momentum p and an energy E to them. These are always related

E = c | p | | (11) |

to each other (where p is understood as a vector in the general case).

To the **energy and** the **Momentum of a photon **to state, we have to fall back on a result of quantum theory (which we also do not justify here): A frequency f can be assigned to a photon ("light particle"). (To put it casually, a light beam with frequency f consists of many photons with frequency f). The frequency f corresponds to a wavelength l (which is connected to c by the equation l f = c). The energy of the photon is now through

E = h f | (12) |

given, where h stands for Planck's constant. The momentum of the photon points in the direction of movement and its magnitude is h / l. Sometimes a photon gets the **Dimensions**m_{Phot} = E / c^{2} º h f / c^{2} assigned. Clearly, however, is among them *Not* the rest mass, but rather the photon version of the dynamic mass.

Photons play an important role in reactions between elementary particles. If the sum of the rest masses after a reaction is smaller than before, additional photons often appear, which carry the missing rest mass as energy (as kinetic energy, if you want). This happens, for example, with nuclear fission. Seen in this way, the main effect of an atomic bomb is to convert rest energy into "light". Sometimes there is also talk of "conversion into energy", but there is no such thing as "pure energy" - concrete particles are always meant, especially photons.

For some purposes it is convenient to express the energy not in terms of speed but in terms of momentum. In non-relativistic mechanics this leads to the formula E = p^{2}/ (2m).

If we try the same thing in relativistic mechanics, that is, with the expressions for the relativistic momentum (formula (4) of the previous section) and the relativistic energy, formula (9), we get an equation that (do the math!) Into the shape

E.^{2} = p^{2} c^{2} + m^{2} c^{4} | (13) |

can be brought. For m = 0 it goes into (11).

The most exciting consequence of this relationship is - somewhat superficially - described in a few words: If it is taken seriously in this form (i.e. without taking the root) and resolved according to E, there is a positive and a negative solution. Already in the twenties of the last century it was concluded from this that a relativistic particle (described by quantum theory) could have states with positive energy and *Negative energy states* to have. The latter are no different from that * Antiparticle* to the type of particle under consideration. This illustrates how profoundly the special theory of relativity has enriched not only our ideas of space and time, but also our knowledge of the matter of which our universe is made.

Addendum: E_{kin} calculate through an integration |

Since we motivated formula (8) above in a relatively cumbersome way, we want to add an elegant derivation in conclusion. The individual steps can be justified more precisely than we do here:

The **kinetic energy** of a body moving with velocity v is the **job**that has to be put into it in order to accelerate it from the state of rest to the speed v. This work is called "force times way", more precisely: than that **Distance integral over the force** to determine. The force, in turn, is that **Derivation of the momentum with respect to time**. Overall, therefore

| (14) |

where the acceleration process between the locations x_{0} and x_{1} runs. The integrand can be transformed as follows:

dx dp / dt = dx dp / you du / dt = you u dp / you, | (15) |

where u stands for the instantaneous speed dx / dt. The integral (14) thus becomes

| (16) |

So we have to differentiate the expression for the relativistic momentum (formula (4) of the previous section) - with u as velocity - according to u, multiply the result by u and integrate the obtained function according to u within the limits of 0 to v. The result is exactly the formula (8) using (9).

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