RELATIVITY

Relativity is the physics of the nature of space and time, a branch of physics that plays a minor role in technology but is crucial in the world of elementary particles.

Overview

At the end of the nineteenth century, the cornerstones of physics were Isaac Newton's mechanics (the theory of forces and motion) and James Clerk Maxwell's electrodynamics. This classical physics referred to locations in space and changes in time and therefore embodied assumptions about the structure of space and time. Among these assumptions was the relationship, or relativity, of measurements. When one reference frame moves with respect to another, different values of some quantities are measured in the two frames. As an example, a hawk might be flying at 80 mph east relative to the ground but 300 mph west relative to an eastbound airplane. The classical conceptions of relativity were too obvious to get much attention: space had a three-dimensional Euclidean geometry, and the flow of time was universal, a clock ticking the same way for all reference frames. Most important, classical space and time were separate.

Questions raised by Maxwell's theory changed all this. The equations of his electrodynamics could only work in a single reference frame. Experimental searches were undertaken for this special reference frame, most notably the 1887 experiment of Albert Michelson and Edward Morley. No trace of such a frame was found; experiments showed Maxwellian electrodynamics to work in all reference frames.

In 1905 Albert Einstein gave the explanation: the classical assumptions about space and time are wrong. Einstein's space-time description, special relativity theory (SRT), gave new relationships of quantities in different reference frames. With these new relationships Maxwellian electrodynamics worked in all reference frames. The mathematics of the new relationships was easy, requiring nothing more sophisticated than a square root. The formulas, the Lorentz transformations for the new relationships, had, in fact, been worked out one year earlier by Hendrik Lorentz, who did not understand their real impact. Though SRT is universally accepted in the scientific community today, its violation of intuition made it revolutionary in 1905. Most revolutionary was the SRT view that space and time were not distinct but had some of the character of different directions in a single space-time entity.

The historical association of SRT and electro-magnetism can be blamed for a common misconception. In SRT there is a fundamental constant of nature, denoted as c , that has the same units as a velocity and has a numerical value very close to 300,000 km/sec. This constant plays an important role in the fundamental structure of space-time. One of its well-known roles is as a velocity limit; the relative velocity of any two objects must have a value less than c . Because light (and certain other) waves propagate at c , it is often called the speed of light. This name, avoided here, too often gives the false impression that SRT is a consequence of observations based on light signals.

The classical conception of space-time, and the Newtonian mechanics that uses it, lasted so long because they were highly accurate for most laboratory and astronomical science. Einsteinian relativistic effects become noticeable only for relative motions at speeds comparable to c , and such speeds are rarely encountered. Even a speed of 1,000 km/hr is only one millionth of c . The situation is opposite for much of elementary particle physics: Speeds are extremely close to c , and special relativistic effects are crucial.

Special Relativity and Elementary Particles

In Newton's mechanics, particles are characterized by an inherent property called mass m . For a particle moving at velocity two other classical quantities are useful, its energy E = ½m?² and its momentum = m? . The usefulness of E and lies in the fact that they are conserved; they do not change in time. For example, suppose you have a set of particles that undergo complicated interactions. Some of the particles could exert forces on each other; some of the particles could join together to form a new particle; some could break apart, forming new particles in another way. Add up all the particle energies at any time, and at any other time, and one gets the same number. In particular the total energy is the same before and after the particle interaction. The same is true for the total momentum.

Since the classical E and are constructed from the velocity , they will have different values in different reference frames. This means, for example, that the total energy in a particular reference frame is different from that in a second frame. This was not a problem in Newton's mechanics. What was important was that in any one reference frame the total E and are conserved.

In SRT, the relativity of is different from that of classical physics. A consequence of this is that if E = ½m?² is conserved in one frame, it is not conserved in another; the same frame-dependent conservation applies to = m . Clearly it is useless, and even meaningless, to have a conservation law that depends on the reference frame. Einstein found that the conservation laws could be saved by changing the definitions of energy and momentum. The SRT definitions, incorporating the Lorentz factorare: If these energy and momenta are conserved in one frame, they are conserved in any other frame.

At least one feature of these equations is immediately surprising: when ? is set to zero, the energy of a particle of mass m is not zero, as in Newtonian mechanics, but rather is given by the famous formula E =mc². A particle has energy even if it is not moving. There exists specialized vocabulary to help sort out the many possible meanings of energy. The value mc² is called the rest energy of a particle, and ?mc² can be called the mass-energy. Often the explicit expression kinetic energy is used to mean the mass-energy minus the rest energy, (? - 1)mc². This kinetic energy has the familiar feature that it is zero for a stationary particle.

It is important to understand that it is the total mass-energy E , not the total kinetic energy, that is the same before and after something happens. A dramatic example of this is the decay of a kind of elementary particle called a pion. Sitting still in a laboratory, a pion can transform itself into two pure bursts of electromagnetic energy, two gamma rays. These bursts have no rest mass, so the death of the pion is a total conversion of rest energy into electromagnetic energy. A more familiar example is nuclear fission. Certain nuclei, such as the isotope uranium-235, spontaneously break into several fragments. The sum of the rest energies of the fragments is less than the rest energy of the original uranium nucleus, so the fragments must have kinetic energy in order for energy to be conserved. This kinetic energy is converted into great amounts of heat in bombs and in reactors.

Two issues of terminology can cause confusion about the meaning of mass. In some older books mass was taken to be what is now denoted as m? . (In potentially confusing contexts the term rest mass is used to denote what we call m .) The second source of confusion is the nearly universal practice of referring to the rest energy of an elementary particle as the particle mass. The rest energy, furthermore, is stated in an unusual system of units: electron volts, or eV (1 kilo electron volt [keV] = 10³ eV, 1 mega electron volt [MeV] = 10⁶ eV, 1 giga electron volt [GeV] = 10⁹ eV, and 1 tera electron volt [TeV] = 10¹² eV). As an example, a proton with a mass of m_p = 1.672610 × 10^-24 grams has a rest energy of m_pc² = 1.503 × 10-3 ergs or 938 MeV. A scientist will speak then of the mass of a proton being 938 MeV and similarly of an electron having a mass of 511 keV.

The practical aspects of other terminology is more obvious. For particles moving at everyday speeds, ? is a useful number, but the Lorentz factor ? is not. For a particle at rest ? is exactly unity, and for a particle moving at 1,000 km/sec the Lorentz factor is unity up to the thirteenth decimal place. For elementary particles produced in accelerators and found in cosmic rays, on the other hand, ? and particle energy are useful, but ? is not. For example, the Large Hadron Collider (LHC), to be built at the European Laboratory for Particle Physics (CERN) near Geneva, will be capable of creating protons with a ? ? 3,700, or equivalently with energy ? × 938 MeV = 3.5 TeV. It is inconvenient to talk of the velocity of such a proton; to eight significant figures it is equal to c .

As an example of the crucial, and surprising, relativistic effects for elementary particles we can imagine a fictitious experiment at the LHC designed to create an exotic new high-mass particle by the joining of two protons in a collision. The left side of Figure 1 shows an appropriate "before" and "after" if one of

FIGURE 1

the protons is a stationary target that is hit by a proton moving at velocity ? . From only the conservation of relativistic energy and momentum it can be shown that the final particle that is formed is moving at a velocity ?? /(1 + ?) and has a mass of For everyday speeds, ? is approximately unity, and the final particle mass and velocity have approximately the classically computed values 2m_p and v /2. For high values of ?, however, the answers are very different. The final particle produced can have a mass much greater than twice m_p and will be moving very nearly at the speed of light.

Though it entails considerable engineering difficulty, the LHC is built to accelerate particles both clockwise and counterclockwise, so that collisions can take place as shown on the right side of the figure, creating an unmoving final particle. The mass of this particle is calculated, from the conservation of relativistic energy and momentum, to be 2?m_p. In the LHC, with ? = 3,700, this is a mass of 7 TeV. For this value of ?, the particle created for a stationary target proton (the left side of the figure) has a mass of 81 GeV, only around 1 percent of 7 TeV. From this it is clear that particles with much larger masses can be created in colliding beam accelerators.

General Relativity

Einstein's 1905 theory is called special relativity because it applies only in the special circumstance that gravitational forces are absent. Around 1915, Einstein successfully extended his space-time ideas to general relativity, a more general theory that can deal with gravitational interactions. This theory is built on the equivalence principle of gravity, the principle that all particles undergo the same acceleration due to gravity. Einstein saw that what was crucial to gravity was the special space-time curves describing how a particle, any particle, could move. For that reason his theory is a geometric one in which the special curves are features of a curved space-time geometry.

Like SRT, general relativity is not crucially important for much of science. Newton's much simpler seventeenth-century theory of gravitational forces is an excellent approximation except for the description of the universe as a whole or for objects like black holes in which gravitational forces are extreme. Unlike SRT, general relativity requires advanced mathematics. Another difference is that general relativity has not been solidly verified to the same extent as SRT. It has, however, passed all experimental tests yet tried.

Ultimately, general relativity cannot be correct; it does not include the quantum effects that scientists believe must be part of every physical interaction. Such a complete theory, combining curved space-time gravity and quantum effects, has been a foremost goal of theoretical physics for more than a half-century. For the past several years some physicists have felt that a new approach called string theory shows promise in providing a path to the complete theory and at the same time resolving longstanding problems in quantum theory.

See also:Einstein, Albert

Bibliography

Mermin, N. D. Space and Time in Special Relativity (McGraw Hill, New York, 1968).

Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation (W. H. Freeman, San Francisco, 1970).

Price, R. H. "A General Relativity Primer." American Journal of Physics50 , 300 (1982).

Taylor, E. F., and Wheeler, J. A. Spacetime Physics: Introduction to Special Relativity, 2nd ed. (W. H. Freeman, San Francisco, 1992).

Thorne, K. S. Black Holes and Time Warps: Einstein's Outrageous Legacy (W. W. Norton, New York, 1994).

Richard H. Price