Mass–energy equivalence
The equivalence of mass and energy (or short: E = mc ²) is the knowledge of relativistic physics, that mass and energy are not independent; rather each physical system with mass has a rest energy
It is the speed of light. This insight was formulated in 1905 by Albert Einstein.
Due to the large factor go energy sales, as it is typical in everyday life, with only imperceptibly small changes in mass associated. Thus, for example the sun, despite their high temperature by only about 0.0001 % of massive, than if it were cold. Therefore, the separation of the concepts of mass and energy remains useful in many areas. In nuclear physics, elementary particle physics and astrophysics, the equivalence of mass and energy occurs, however, far more apparent. Meeting, for example, a particle and the corresponding antiparticle each other, they annihilate each other. The energy of the resulting photon corresponds to the mass of the particles thereby destroyed. But even atomic nuclei are less massive than their unbound nucleons due to the binding energy of almost 1%.
Overview and Examples
The fact that the equivalence of mass and energy in classical physics and in everyday life unnoticed, can be understood from the size of the factor out. After the energy conversions of normal size correspond ( approximately in chemical reactions such as combustion or generation of heat by mechanical work ) only extremely small changes in the mass who are familiar with the scale today hardly measurable. As a result, two separate conservation laws have been established for closed systems: conservation of the total mass, conservation of total energy. However, since conversions between kinetic energy and rest energy (examples: inelastic impact, radioactive decay ) are possible and only the rest energy contributes to the mass, the mass conservation is not generally valid. The energy transfer associated with a change in the mass of an object is also called the mass gain or mass defect. Instead of two conservation laws so you have only one, the energy conservation law.
In everyday situations, read the rest mass energy of a body exceeds its kinetic energy by many orders of magnitude. Even the kinetic energy of a satellite in Earth orbit (which is after all so great that it can burn up the satellite when it is transformed by re-entry of the satellite into the atmosphere in an equal quantity of heat ) is less than a billionth of its rest energy:
A hydrogen atom, - consisting of an electron and a proton - has together about 1/ 70 million less mass than the two free particles. This mass difference has become available as the binding energy in the formation of the atom. For nuclei of this mass defect is even quite large: for example, about 0.8% at 12C.
Known examples of the equivalence of mass and energy are:
- Annihilation radiation: A particle pair electron-positron, which together have a mass of about, can dissolve in massless radiation: two gamma quanta of each 511 keV energy. The rest energy of the system before the mutual destruction is exactly as large as the energy of the emerging radiation.
- Nuclear fission: a nucleus of the element uranium can burst into a plurality of fragments, the masses are together about 0.1 % smaller than the initial uranium core. The liberated energy corresponds exactly according to this decrease in mass and can (with cleavage of a corresponding amount of substance ), inter alia, as Explosion ( Atomic Bomb ) or heat source ( nuclear power plant ) appear.
- The sun loses solely by radiated from her light every second around 4 million tons of mass. Compared with the total mass of the sun around but this effect is negligible. Even after several billion years, the sun has lost in this way is far less than one-thousandth of its mass.
Classification
Modern physics formulates the concepts of mass and energy by using the energy - momentum relation of special relativity: It follows that ( hereinafter referred to as "body" ) of each completed physical system a total energy and a pulse whose values may be different depending on the chosen reference system, and a constant mass, which is independent of the reference system property of the body. The sizes are the four components of the energy-momentum four-vector of the body. The standard of this four-vector is determined (up to a constant factor c ) by the mass:
According to the energy converted:
In the center of mass system () is obtained for the energy again, often referred to as rest energy.
From another reference system from the same body has different values for the four components obtained by conversion using the Lorentz transformation. Moves the reference system with speed against the body, he has the speed in this reference system, and his energy and momentum are determined in accordance with:
The norm of the four-vector is maintained ( see above), the mass is thus a Lorentzinvariante, formerly known as rest mass.
When one develops the equation in powers of in a Taylor series, we obtain:
The " zeroth " element of this series is again the rest energy of the body. All higher terms together form the kinetic energy. In the first of these terms is lifted out and the result is the classical kinetic energy
This is a good approximation if the nonrelativistic case, all other elements may be ignored (i.e., ), because of magnitude of included. At very high speeds, these higher terms can not be neglected. Then represent the accelerating growth of the kinetic energy of relativistic velocities.
Gravity
Einstein extended in 1907 his ideas on gravitation. The equivalence principle, ie the equality of inertial and gravitational mass, led him to the conclusion that an increase in the rest energy of a system is also an increase in the heavy mass result. The continuation of this idea in the context of the general theory of relativity showed that the energy -momentum tensor is to be regarded as the source of the gravitational field.
An example is the gravitational collapse. If the interior of a star goes nuclear heat generation, his matter is concentrated in such a small space that at sufficiently large mass, the ever-increasing gravitational field itself contributes through its field energy to further attraction and contraction. The result is a black hole.
Relativistic mass
In older textbooks the equivalence formula is often related not only to the rest energy and the invariant mass, but also to the relativistic total energy and thus a so-called relativistic mass or dynamic:
According to this notation are energy and ( relativistic ) mass under all circumstances equivalent. This interpretation of the mass, however, is questionable (see the section on so-called relativistic mass ), and Einstein himself rejected them. In many modern textbooks, therefore, the concept of relativistic mass is rejected as impractical. Instead, the equivalence relation should be used exclusively in connection with the Lorentz invariant mass and the rest mass energy.
History
Overview
The relationship between mass, energy, and speed of light was considered electrodynamics as early as 1880 by various authors in the context of Maxwell. Joseph John Thomson ( 1881), George Searle (1897 ), Wilhelm Wien (1900 ), Max Abraham (1902 ) and Hendrik Lorentz (1904 ) developed, that the electromagnetic energy of the body adds an " electromagnetic mass " according to the formula ( in modern notation)
At the same formula came Hasenohrl Friedrich ( 1904/5 ) by considering the electromagnetic cavity radiation of a body, he noted the dependence of the mass on the temperature. Henri Poincaré (1900 ), however, concluded from observations on the reaction principle that electromagnetic energy of a " fictitious" mass of
Corresponds. The electromagnetic mass was often referred to as "apparent" mass since a distinction this time being " true" from the mechanical mass Newton.
Albert Einstein introduced in 1905 from the recently he developed special relativity theory that the mass of a body has to change to when the body absorbs the energy or emit. He won this result for the case that there are electromagnetic radiation in energy sales. But when he first recognized the general validity: The equivalence must also apply to all other possible forms of energy revenues, and beyond mutandis, the total rest energy and the total mass
This was the equivalent of a comprehensive theory of special relativity, embedded. All previous speculation about the electromagnetic nature of the ground had pointed in the wrong direction, because in the special theory of relativity applies without exception the equivalence of mass and rest energy, regardless of whether the mass of electromagnetic origin or not.
This equivalence was originally called the " inertia of energy ," since they each form of energy -. Including the kinetic energy of the center of mass motion of the whole system - an inert mass attributed to the so-called relativistic mass Such a use of words, however, as relativistic mass in the article explains misleading, because the inertia of a fast-moving system depends on its direction of motion.
There followed a number of other theoretical derivations of equivalence in the mold under a variety of conditions (see below the time table ). Einstein himself has published 18 such derivations, the last in 1946. Regularly highlighted that with the equivalence was not already shown in the form, but only in the form with any constant term. Since such a summand is always free because when specifying a total energy of zero is for the convention, you can (1907 ) as " much more natural " choice (Einstein) set equal to zero it. In this form, the equivalence of mass and resting energy already an established part of theoretical physics was before it could be verified by measurements.
Experimentally, the equivalence in the form of 1920 high available using the mass defect of the nuclear masses. Quantitatively, it was confirmed from the 1930s with nuclear reactions in which both energy sales and the difference of the masses of the reactants were before and after the reaction measurable. Initially, the error limits were, however, at 20 %.
An experimental examination of equivalence in the mold is possible through measurement of energy conversion in the generation or destruction of particles. Such processes were first published in 1934 by Enrico Fermi theoretically considered to explain the formation of the beta rays. Fermi took quantum Dirac equations for the electrons thus generated. This is based on the energy - momentum relation of special relativity and thus writes the generation of the electron at rest () the energy consumption. This was confirmed by measuring the maximum kinetic energy of the electrons and comparison with the energy balance of nuclear transformation.
The creation of an electron -positron pair and their mutual destruction were observed since 1932. The measured energy revenues correspond to those which are to be expected on the basis of the equation. These processes, however, were not interpreted until the 1940s as creation or annihilation of particles with rest mass. Instead, she pointed as a suggestion of an existing in a Dirac sea electron from a state of negative energy into a positive energy state, with constant mass. Only with the interpretation of particles and antiparticles as each separate class of particles the image from the Dirac sea was unnecessary, and the energy expenditure of the creation and annihilation of particles and antiparticles was the experimental confirmation of the equation.
Today, the validity of the equivalence of mass and energy is experimentally confirmed with very high accuracy:
Chronology
Starting with 1905 Interpretation and meaning of the equivalence of mass and energy were gradually developed and deepened.
Einstein's derivation
Einstein came in 1905 by the following thought experiment on the relationship between mass and energy. A similar thought experiment was developed in 1900, but can not resolve satisfactorily Poincaré.
In a reference system at rest, a body and has a certain rest energy, which we need to know anything further with. He sends two identical flashes of light of equal energy in opposite direction. Then, the pulses of the light flashes of equal size, but opposite, so that the body remains at rest due to the conservation of the total momentum. Because of the conservation of energy, the body now has the energy
We consider the same process from a second frame of reference which is moving relative to the first speed of the emission direction of the light flashes. The values of all calculated in the second system energies are denoted by. It could be that the energy scales of both reference systems have different zero points, which differ by a constant. Since the energy conservation in the second reference system as well as in the first applies ( principle of relativity ), it follows
Since the body in the first system remains at rest, it moves in the second system after the issue with the same speed as before. Therefore its energy is the kinetic energy is greater than the first in the second reference system. Therefore:
By subtracting these two pairs of equations in pairs from each other, fall out the unknown rest energies and the constant and one obtains:
The key point is: The two light flashes that have opposite direction and equal energy in the first reference system, are also opposes the second reference system because of the choice of direction but have different energies. One shows redshift, the other blue shift. After the Lorentz transformation of the electrodynamic fields are their energies or where. Together their energy is thus greater than the first reference system:
The two values for the kinetic energy before and after the emission are therefore also different according to the above equation. By issuing the kinetic energy decreases by
Since the emission, the velocity of the body remains the same, but after that he has less kinetic energy than before, its mass must have decreased. To determine this change, we use the formula valid in the limit and develop the right side of the last equation in powers up to the member. There arises. So the output of the power leads to a decrease in mass.
Einstein closes this 1905 published consideration with the words (symbols modernized ):
Einstein circumvents the problem of the unknown rest energy by this size can be eliminated from the equations in his thought experiment. For the energy output it chooses electromagnetic radiation and derives the change in the crowd. In 1905, he adds without proof of the statement that this applied energy loss for each species. From 1907/8, he suggests, "we can dispose of the zero point [ ... ] [... ] to be construed as any inert mass supply of energy ", ie.
E = mc ² and the Atomic Bomb
Ionising radiation Henri Becquerel, Marie and Pierre Curie and Ernest Rutherford had observed in 1897 that nuclear reactions are millions of times more energetic than chemical reactions. The energy source of Rutherford and Frederick Soddy (1903 ) was a in the bodies befindliches, enormous reservoir suspected of latent energy, which must be present in normal matter. Rutherford ( 1904) speculated that it might control the decay of radioactive elements and one day from a small amount of matter an enormous amount of energy could release. With Einstein's equation ( 1905), one could read this energy to the different core masses, which could actually be detected in the 1930s.
However, the equation does not say how to set the fission of heavy nuclei in transition. The decisive factor was the observation of the induced nuclear fission by Otto Hahn and Fritz Strassmann, and that these released neutrons can trigger a chain reaction in uranium enriched. Unlike assert popular science reports, hence the connection between rest energy and mass in the development of the atomic bomb ( " Manhattan Project " in the U.S. from 1942) played no special role. Albert Einstein influenced the development of the atomic bomb than by his physical findings, but at best politically, namely, by his letter to President Roosevelt in which he advocated the development of the atomic bomb by the Americans.