Special relativity / Elementary Tour part 6: E=mc²

Now that the new features of space and time are sufficiently explored, it’s time to examine how this affects the way objects move, either freely or when influenced by forces. In other words: How does relativistic mechanics work? This question has led to the prediction (and, eventually, the measurement) of further relativistic effects, such as the following. In pre-Einstein physics, the ratio of the strength of a force acting on a solid object and the resulting change in the object’s velocity (its acceleration) was thought to be constant. Physicists called it the object’s mass (more precisely: its inertial mass).

In special relativity, however, the inertial mass of a body directly depends on its speed – the higher the speed, the higher the mass. This effect is crucial for anyone operating a high-energy particle accelerator in which elementary particles are accelerated to speeds near that of light. The following image shows one such accelerator, the VUV ring at Brookhaven National Laboratory, with a circumference of 51 metres:

The machine is used to produce what is called synchrotron radiation – a form of electromagnetic radiation that is especially concentrated and intensive, and that can be used for a variety of investigations from basic physics to materials science to medicine. Inside this accelerator, electrons are racing around at slightly more than 99.9999 per cent of the speed of light. If one were to ignore the relativistic mass increase, one couldn’t even get this machine to run.

The relativistic increase of mass happens in a way that makes it impossible to accelerate an object to light speed: The faster the object already is, the more difficult any further acceleration becomes. The closer the object’s speed is to light speed, the greater the increase in inertial mass; to reach light speed exactly would require an infinitely strong force acting on the body. This enforces special relativity’s speed limit: No material object can be accelerated to light speed.

The increase in inertial mass is part of a more general phenomenon, the relativistic equivalence of mass and energy: If one adds energy to a body, one automatically increases its mass; if one takes energy away from it, one decreases its mass. In the case of acceleration, the object in question gains kinetic energy (“movement energy”), and this increase in energy automatically means an increase in mass.

On the other hand, even an object at rest turns out to have a certain amount of energy. Energy and (inertial) mass are inextricably linked by Einstein’s famous formula. Every body of mass m will necessarily have a total energy

E = mc²

(where the constant c is the speed of light). Inverting the formula, every body which has the total energy E will have an inertial mass m = E/c². Although physicists used to measure them in different units (which necessitate the conversion factor c²), mass and energy really are the same. The only reason pre-Einstein physicists defined them as separate concepts in the first place is because they didn’t yet know the whole picture.

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Definitioner

Brookhaven National Laboratory

Nationales Forschungszentrum in den Vereinigten Staaten mit Hauptsitz Long Island, New York. Das BNL betreibt unter anderem den Relativistischen Schwerionenbeschleuniger RHIC (Relativistic Heavy Ion Collider), mit dessen Hilfe sich extreme Materiezustände präparieren lassen, wie sie Sekundenbruchteile nach dem Urknall im frühen Universum herrschten. Außerdem betreibt das BNL Beschleuniger zur Erzeugung von Synchrotronstrahlung, von denen einer, der VUV-Ring, auf der Seite E=mc2 im Kapitel Spezielle Relativitätstheorie von Einstein für Einsteiger abgebildet ist. Webseiten des Brookhaven National Laboratory National Synchrotron Light Source II am Brookhaven National Laboratory

velocity

In physics, velocity is a combination of two aspects: First of all, how fast an object is, in other words: how long a distance it moves in a given time (its "speed"). Secondly, the direction in which an object moves. Physicists combine these two informations into a single mathematical object, called a "vector", and this is what is called the velocity. For instance, when a car goes around a curve with 100 miles per hour, its speed is constant, but as it changes its direction of movement, its velocity changes correspondingly.

time

It is a fact of life that not all events in our universe happen concurrently - instead, there is a certain order. Defining a time coordinate or defining time, the way physicists do it, is to define a prescription to associate with each event a number so as to reflect that order - if event B happens after event A, then the number associated with B should be larger than that associated with A. The first step of this definition is to construct a clock: Choose a simple process that repeats regularly. (What is "regular"? Luckily, in our universe, all elementary processes such as a swinging pendulum, the oscillations of atoms or of electronic circuits lead to the same concept of regularity.) As a second step, install a counter: A mechanism that, with every repetition of the chosen process, raises the count by one. With this definition, one can at least assign a time (the numerical value of the counter) to events happening at location of the clock. For events at different locations, an additional definition is necessary: One needs to define simultaneity. After all, the statement that some far-away event A happens at 12 o'clock is the same as saying that event A and "our clock counter shows 12:00:00" are simultaneous. The how and why of defining simultaneity - a centre-piece of Einstein's special theory of relativity - are described in the spotlight topic Defining "now". With all these preparations, physicists can, in principle, assign a time coordinate value ("a time") to any possible events, and describes how fast or how slow processes happen, compared to that time coordinate.

strong force

One of the four fundamental forces in our universe (the others are electromagnetism, weak nuclear force and gravity). The strong force binds the quarks to form compound particles such as protons and neutrons. Indirectly, it is also responsible for holding together protons and neutrons in atomic nuclei.

speed

An object's average speed is the distance it moves during a given period of time, divided by the length of the time interval. If you make the time interval infinitely small, the result is the object's speed at one particular moment in time. The notion of speed can be applied to waves in different ways; for instance, for a simple wave, the phase speed is the speed at which any given wave crest or wave propagates through space. See also the more general entry velocity.

speed of light

The speed at which light or, more generally, electromagnetic radiation propagates through space (especially: through empty space). Central quantity in special relativity: There, the constancy of the speed of light is a basic postulate: every observer (more precisely: every inertial observer) that measures the speed of light in vacuum obtains the same constant value, c=299,792,458 metres per second. Another important relativistic aspect of the speed of light is that it defines an absolute upper speed limit: In special relativity, nothing can move faster than light, and information or influence at most be transmitted at light-speed. In general relativity, the same law is in force locally: No object, no matter, no information can directly overtake or catch up with light (cf. causality). Basic information about the role of light speed in special relativity can be found in the chapter Special relativity of Elementary Einstein.

special relativity

Albert Einstein's theory of the fundamentals of space, time, and movement (but not gravity). For a brief introduction, check out the chapter Special relativity of Elementary Einstein. Selected aspects of special relativity are described in the category Special relativity of our Spotlights on relativity.

space

In a strict sense: Space as we know it from everyday life: the totality of all locations in which objects can sit, with three dimensions. In a more general sense used by mathematicians, all kinds of sets of points are spaces - a line for instance, which has but a single dimension, or a two-dimensional surface, but also higher-dimensional spaces. Also, in such more general spaces, geometry can be different from the standard Euclidean geometry taught in high schools - such spaces can be curved.

relativistic

Models, effects or phenomena in which special relativity or general relativity play a crucial role are called relativistic. Examples are relativistic quantum field theories as theories based on special relativity, or the relativistic perihelion shift as a consequence of general relativity. In addition, conditions under which the difference between relativistic physics and ordinary, classical physics are especially pronounced, are also called relativistic. For instance, when material objects reach speeds close to speed of light, one talks of relativistic speeds, while speeds that are so small compared to light as to make relativistic effects undetectably small are non-relativistic.

relativistic mass

One prediction of special relativity is that, the faster an object already is, the more difficult it is to accelerate it even further. One consequence of this is that it is impossible to accelerate a material object to the speed of light: The faster the object already is, the more force has to be used to increase its speed, and close to the speed of light, this effect becomes so strong that, finally, one would have to use infinite force to effect the final, decisive acceleration.

Traditionally, in classical physics, the resistance of an object to changes of its state of motion is its (inertial) mass. The relativistic mass of an object is defined in the same way, and the value an observer measures for this relativistic mass increases as an object moves faster and faster relative to that observer.

relativistic (perihelion advance, relativistic)

For planetary orbits, there is a small difference between the predictions of Newtonian gravity and general relativity. For instance, in Newton's theory, the orbital curve of a lonely planet orbiting a star is an ellipse. In general relativity, it is a kind of rose or rhodonea curve. Such a curve is similar to an ellipse curve, which shifts a bit with each additional orbit. The shift can be defined by looking at the point which is closest to the sun (perihelion) on each orbit. The additional relativistic shift is, hence, called relativistic perihelion shift or relativistic perihelion advance. A picture can be seen on the page A planet goes astray in the chapter General relativity of Elementary Einstein.

particle accelerator

The most important experimental technique of particle physics: accelerating electrically charged particles with the help of electric forces, make them collide with each other and, from the result of the collision, draw conclusions about the properties of elementary particles and their interactions. It is an intriguing possibility, suggested by models based on the ideas of string theory, that particle accelerators such as the LHC might actually produce miniature black holes (for more about this, see the spotlight text Particle accelerators as black hole factories?).

mechanics

Branch of physics dealing with the motions of objects and how they react to forces acting on them. Depending on the framework used, there is classical mechanics, relativistic mechanics and quantum mechanics.

relativistic mechanics (mechanics, relativistic)

The generalization of classical mechanics that takes into account the effects of special relativity. The basic laws are almost unchanged: First of all, bodies on which no external forces act stay at rest or move with constant speed along straight paths - in the language of special relativity: such bodies move on straight lines in spacetime. Secondly: The total force acting on a body is equal to the change of its momentum over time (but notice: this momentum is defined using the body's relativistic mass, which depends on the bodies speed relative to the observer). Thirdly, mass and momentum are conserved quantities - their total sum is the same whenever particles interact (this is equivalent to a slightly modified version of the "action equals reaction" principle of classical mechanics). There exists an elegant reformulation of these laws of mechanics using four-dimensional concepts adapted to the geometry of spacetime, such as the "four-momentum".

mass

In classical physics, mass plays a triple role. First of all, it is a measure for how easy it is to influence the motion of a body. Imagine that you're drifting in emtpy space. Drifting by are an elephant and a mouse, and you give each of them a push of equal strength. The fact that the mouse abruptly changes its path, while the elephant's course is as good as unaltered, is a sure sign that the mass (or, in the language of physics, the inertia or inertial mass) of the elephant is much greater than that of the mouse. Secondly, mass is a measure of how many atoms there are in a body, and of what type they are. All atoms of one and the same type have the same mass, and adding up all those tiny component masses, the total mass of the body results. Thirdly, in Newton's theory of gravity, mass determines how strongly a body attracts other bodies via the gravitational force, and how strongly these bodies attract it (in this sense, mass is the charge associated with the gravitational force). In special relativity, one can also define a mass that is a measure for a bodies resistance to changing its motion. However, the value of this relativistic mass depends on the relative motion of the body and the observer. The relativistic mass is the "m" in Einstein's famous E=mc² (cf. equivalence of mass and energy). The relativistic mass has a minimum for an observer that is at rest relative to the body in question. This value is the so-called rest mass of the body, and when particle physicists talk of mass, this is usually what they mean. Just as in classical physics, the rest mass is a kind of measure for how much matter the body is made up of - with one caveat: For composite bodies, the energies associated with the forces holding the body together contribute to the total mass, as well (another consequence of the equivalence of mass and energy). In general relativity, mass still plays a role as a source of gravity; however, it has been joined by physical quantities such as energy, momentum and pressure.

light

Light in the strict sense of the word is electromagnetic radiation the human eye can detect, with wave-lengths between 400 and 700 nano metres. In relativity theory and in astronomy, the word is often used in a more general sense, encompassing all kinds of electromagnetic radiation. For instance, astronomers might talk about "infrared light" or "gamma light"; in this context, light in the stricter sense is referred to as "visible light". Within classical physics, the properties of light are governed by Maxwell's equations; in quantum physics, it turns out that light is a stream of energy packets called light quanta or photons. In the context of relativistic physics, light is of great interest, and for a number of reasons. First of all, the speed of light plays a central role in both special and general relativity. Also, there are a number of interesting effects in general relativity which are associated with the propagation of light, namely deflection, the Shapiro effect and the gravitational redshift.

light speed

See speed of light.

kinetic energy

A type of energy that has to be ascribed to an object simply because that object moves relative to the reference frame. In classical, pre-Einstein physics, the amount of energy is given by one half, times an object's mass, times the square of its speed.

force

In mechanics: Influence acting on a body, trying to accelerate it.

More generally: All influences by which elementary or other particles can interact; in this sense, force and interaction are synonymous. In the standard model of particle physics, there are three elementary forces: electromagnetism, the weak (nuclear) force and the strong (nuclear) force, while there is no quantum description of the fourth fundamental interaction, gravity.

energy

Physical quantity with the special property that, in physical processes, energy is neither destroyed nor created, simply transformed from one form of energy to another. Some of the different forms of energy that are defined separately are kinetic energy, thermal energy and the energy carried by electromagnetic radiation. Processes that transform one form of energy into another take place in all machines we use in everyday life, from the engine of a subway train (electrical energy into kinetic energy of the train) to an electric blanket (electrical energy into thermal energy). One important consequence of special relativity is that energy and mass are completely equivalent - two different ways to define what is, on closer inspection, one and the same physical quantity. See the keyword equivalence between mass and energy.

elementary particles (elementary particle)

Within the commonly accepted models of physics, certain particles are not built of even more fundamental sub-particles - they are themselves elementary. Examples for elementary particles are electrons, quarks and neutrinos, while particles such as protons or neutrons consist of sub-units and are thus not elementary. The study of elementary particles is called particle physics, under which keyword some more information about the theoretical framework of elementary particles can be found.

acceleration

Every change of velocity with time is an acceleration.

This definition is slightly different from our everyday usage of the word. Ordinarily, we talk of an object accelerating when it becomes faster and faster. The physics definition covers two more situations. An object that decelerates, becomes slower, thus changes its velocity and, in the physics sense, undergoes a (negative) acceleration. Also, in physics, velocity is not the same as speed. A constant velocity implies not only constant speed, but also a constant direction of movement. Once the direction changes, so does the velocity - the change in velocity is associated with the change in the direction of movement. Thus, in the physics sense, even a car going around a curve of the road at constant speed undergoes acceleration.

Special relativity / Elementary Tour part 6: E=mc²

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