Wikipedia:Mass

From Global Warming Art

Sir Isaac Newton (1642-1727)

Mass (from Greek μάζα) is a concept used in the physical sciences to explain a number of observable behaviours, and in everyday usage, it is common to identify mass with those resulting behaviors. In particular, mass is commonly identified with weight. But according to our modern scientific understanding, the weight of an object results from the interaction of its mass with a gravitational field, so while mass is part of the explanation of weight, it is not the complete explanation.

For example, a mail carrier lifting a heavy package on earth may associate the heaviness (weight) of the package with the mass of its contents. This is a reasonable association for objects on earth. However, if the same package were on the moon it would weigh much less and would be easy to lift. Therefore, the mass of a package is only part of the reason that the package is difficult to lift on earth. The complete reason involves the interaction of the package’s mass with the gravity of the earth.

Also, a groundskeeper encountering two large rocks may associate the size of the rocks with their respective masses. And from this association the groundskeeper may expect the larger rock to be heavier and more difficult to move. However, if the larger rock were composed of pumice and the smaller of granite, then the smaller rock may in fact be much heavier. Mass is part of the explanation of an object’s size but not the complete explanation. The complete explanation involves mass, structure, and composition.

The human body is equipped with physical senses through which one can experience many of the effects associated with mass. One can visually observe an object to determine its size, lift it to feel its weight, and push it to feel the force of its inertial resistance to changing motion. These human experiences are all part of our modern understanding of mass, but none completely epitomizes the abstract concept of mass. The abstract concept did not come from a specific type of human experience. Rather, it came from a synthesis of many different types of human experience.

The modern concept was introduced in, and is central to, Isaac Newton’s explanation of gravitation and inertia. Prior to Newton’s time, the various gravitational and inertial phenomena were viewed as distinct and potentially unrelated. However, Isaac Newton united these phenomena by asserting that they all stemmed from a single underlying property called mass. Since Newton’s time, this abstract concept of mass has grown to include explanations for both quantum and relativistic effects. (See the following section entitled “Summary of concepts of mass” for a brief summary of mass-related phenomena.)

Units of mass

The primary instrument used to measure mass is the scale or balance scale. In the SI (system of units), mass is measured in kilograms, kg. Many other units of mass are also employed, such as:

• gram: 1 g = 0.001 kg (1000 g = 1 kg)
• tonne: 1 tonne = 1000 kg
• MeV/c² (Typically used to specify the mass of subatomic particles. See mass-energy equivalence)

Outside the SI system, a variety of different mass units are used, depending on context, such as the:

• Slug
• Pound
• atomic mass unit
• Planck mass
• solar mass

In normal situations, the weight of an object is proportional to its mass, which usually makes it unproblematic to use the same unit for both concepts. However, the distinction between mass and weight becomes important for measurements with a precision better than a few percent (due to slight differences in the strength of the Earth's gravitational field at different places), and for places far from the surface of the Earth, such as in space or on other planets.

Because of the relativistic connection between mass and energy (see mass in special relativity), it is possible to use any unit of energy as a unit of mass instead. For example, the eV energy unit is normally used as a unit of mass (roughly 1.783×10⁻³⁶ kg) in particle physics. A mass can sometimes also be expressed in terms of length. Here one identifies the mass of a particle with its inverse Compton wavelength (1 cm⁻¹ ≈ 3.52×10⁻⁴¹ kg).

Summary of concepts of mass

The above diagram illustrates five interrelated properties of mass together with the proportionality constants that relate these properties. Every sample of mass is believed to exhibit all five properties, however, due to extremely large proportionality constants, it is generally impossible to verify more than two or three properties for a specific sample of mass.

• The Schwarzschild radius (r_s) represents the ability of mass to cause curvature in space and time.
• The standard gravitational parameter (μ) represents the ability of a massive body to exert Newtonian gravitational forces on other bodies.
• Inertial mass (m) represents the Newtonian response of mass to forces.
• Rest energy (E₀) represents the ability of mass to be converted into other forms of energy.
• The Compton wavelength (λ) represents the quantum response of mass to local geometry.

In physical science, one may distinguish conceptually between at least seven attributes of mass, or seven physical phenomena that can be explained using the concept of mass:^[1]

• Inertial mass is a measure of an object's resistance to changing its state of motion when a force is applied. It is determined by applying a force to an object and measuring the acceleration that results from that force. An object with small inertial mass will accelerate more than an object with large inertial mass when acted upon by the same force. One says the body of greater mass has greater inertia.

• The amount of matter in certain types of samples can be exactly determined through electrodeposition or other precise processes. The mass of an exact sample is determined in part by the number and type of atoms or molecules it contains, and in part by the energy involved in binding it together (which contributes a negative "missing mass," or mass deficit).

• Active gravitational mass is a measure of the strength of an object’s gravitational flux (gravitational flux is equal to the surface integral of gravitational field over an enclosing surface). Gravitational field can be measured by allowing a small ‘test object’ to freely fall and measuring its free-fall acceleration. For example, an object in free-fall near the Moon will experience less gravitational field, and hence accelerate slower than the same object would if it were in free-fall near the earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass.

• Passive gravitational mass is a measure of the strength of an object's interaction with a gravitational field. Passive gravitational mass is determined by dividing an object’s weight by its free-fall acceleration. Two objects within the same gravitational field will experience the same acceleration; however, the object with a smaller passive gravitational mass will experience a smaller force (less weight) than the object with a larger passive gravitational mass.

• Energy also has mass according to the principle of mass–energy equivalence. This equivalence is exemplified in a large number of physical processes including pair production, nuclear fusion, and the gravitational bending of light. Pair production and nuclear fusion are processes through which measurable amounts of mass and energy are converted into each other. In the gravitational bending of light, photons of pure energy are shown to exhibit a behavior similar to passive gravitational mass.

• Curvature of spacetime is a relativistic manifestation of the existence of mass. Curvature is extremely weak and difficult to measure. For this reason, curvature wasn’t discovered until after it was predicted by Einstein’s theory of general relativity. Extremely precise atomic clocks on the surface of the earth, for example, are found to measure less time (run slower) than similar clocks in space. This difference in elapsed time is a form of curvature called gravitational time dilation. Other forms of curvature have been measured using the Gravity Probe B satellite.

• Quantum mass manifests itself as a difference between an object’s quantum frequency and its wave number. In natural units: m² = ω² − k². The quantum mass of an electron, the Compton wavelength, can be determined through various forms of spectroscopy and is closely related to the Rydberg constant, the Bohr radius, and the classical electron radius. The quantum mass of larger objects can be directly measured using a watt balance.

Inertial mass, passive and active gravitational mass, and the various other mass-related phenomena are conceptually distinct. However, every experiment to date has shown these values to be proportional, and this proportionality gives rise to the abstract concept of mass. If, in some future experiment, one of the mass-related phenomena is shown to not be proportional to the others, then that specific phenomena will no longer be considered a part of the abstract concept of mass.

Weight and amount

Anubis weighing the heart of Ani, 1285 BC

The concepts of passive gravitational mass and atomic mass grew out of the much older concepts of weight and amount. In the Egyptian religious illustration to the right, Anubis is using a balance scale to weigh the heart of Ani. A balance scale balances the force of one object’s weight against the force of another object’s weight. Weight, by definition, is a measure of the force that an object experiences when its mass interacts with a gravitational field.

The gravitational field of the earth is nearly uniform at all locations on the earth’s surface; therefore, an object’s weight changes only slightly when it is moved from one location to another, and these changes went unnoticed through much of history. When two objects are near each other, they experience similar gravitational fields; hence, if their weights are identical then their masses must also be identical. The two sides of a balance scale, for example, are close enough that the scale, by comparing weights, is also understood to compare passive gravitational masses.

The concept of amount is very old and predates recorded history, so any description of the early development of this concept is speculative in nature. However, one might reasonably assume that humans, at some early era, realized that the weight of a collection of similar objects was proportional to the number of objects in the collection. Consequently, historical weight standards were often defined in terms of amounts. The Romans, for example, used the Carob seed (carat or siliqua) as a measurement standard. If an object’s weight was found to be equivalent to 1728 Carob seeds, then the object was said to weigh one Roman pound. The Roman ounce (uncia) was a twelfth of a pound, or 144 Carob seeds.

The British also used the pound as a weight standard in their measurement system. However, the meaning of the pound had become distorted over time, and a number of different definitions were used within the British isles. Amongst these were the avoirdupois pound and the now obsolete tower, merchant's and London pounds.^[2] As shown in the table, the various British pounds were related through simple fractions. The avoirdupois pound, for example, was equal to thirty five twenty sevenths of a tower pound. This meant that if twenty seven avoirdupois pounds were placed on one side of a balance scale and thirty five tower pounds were placed on the other side, the two sides would balance.

The fact that the British pounds were related through simple fractions indicates that they had common denominators. The least common denominator (LCD) is obtained in the table by calculating the least common multiple of the denominators along each column, or equivalently, the least common multiple of the numerators along each row. The numerators along the first row are 175, 35, 28, and 35, the least common multiple being 700. The least common multiples of the numerators in the second, third, fourth, and fifth rows are respectively 576, 540, 675, 720. The existence of common denominators means that the pounds of the British system can be defined as amounts. As shown in the table, the avoirdupois pound is equal to 700 LCD units, the Troy pound is equal to 576 LCD units, and so forth. The British, however, didn’t use the LCD in their weight system, but rather, they used the barley grain mass which was equal to one tenth of the LCD. The avoirdupois pound was thus defined to be 7,000 barley grains.

Various atoms and molecules as depicted in John Dalton's A New System of Chemical Philosophy (1808).

A stylised lithium atom

The name atom comes from the Greek ἄτομος/átomos, α-τεμνω, which means uncuttable, something that cannot be divided further. The philosophical concept that matter might be composed of discrete units that cannot be further divided has been around for millennia. However, empirical proof and the universally acceptance of the existence of atoms didn’t occur until the early 1900’s.

As the science of chemistry matured, experimental evidence for the existence of atoms came from the law of multiple proportions. When two or more elements are combined to form a compound, their masses are always in a fixed and definite ratio. For example, the mass ratio of nitrogen to oxygen in nitric oxide is seven eights. Ammonia has a hydrogen to nitrogen mass ratio of three fourteenths. The fact that elemental masses always combined in simple fractions implies that a least common denominator for elemental mass must exist.

In principle, one should be able to place the elemental mass ratios that occur in nature in a single table analogous to the above table for British pounds. And furthermore, one should be able to go through that table using a process somewhat analogous to the one used above for British pounds, and determine the least common denominator of elemental mass. In 1805, the chemist John Dalton published his first table of relative atomic weights, listing six elements, hydrogen, oxygen, nitrogen, carbon, sulfur, and phosphorus, and assigning hydrogen an atomic weight of 1. And in 1815, the chemist William Prout concluded that the mass of the hydrogen atom was in fact the least common denominator of all of the elemental masses.

If Prout's hypothesis had proven accurate, then the abstract concept of mass, as we now know it, might have never evolved, since mass could always be defined in terms of amounts of the hydrogen atomic mass. Prout’s hypothesis; however, was found to be inaccurate in two major respects. First, further scientific advancements revealed the existence of smaller particles, such as electrons and quarks, whose masses are not related through simple fractions. And second, the elemental masses themselves were found to not be exact multiples of the hydrogen atom mass, but rather, they were near multiples. Einstein’s theory of relativity explained that when protons and neutrons come together to form an atomic nucleus, some of the mass of the nucleus is released in the form of binding energy. The more tightly bound the nucleus, the more energy is lost during formation and this binding energy loss causes the elemental masses to not be related through simple fractions.

Hydrogen, for example, with a single proton, has an atomic weight of 1.007825 u. The most abundant isotope of iron has 26 protons and 30 neutrons, so one might expect its atomic weight to be 56 times that of the hydrogen atom, but in fact, its atomic weight is only 55.9383 u, which is clearly not an integer multiple of 1.007825. Prout’s hypothesis was proven inaccurate in many respects, but the abstract concepts of atomic mass and amount continue to play an influential role, and the atomic mass unit continues to be the unit of choice for very small mass measurements. (Although, the definition of the atomic mass unit is no longer tied to the hydrogen atom).

When the French invented the metric system in the late 1700s, they also used an amount to define their mass unit. The gram was originally defined to be equal in mass to the amount of pure water contained in a one milliliter container. This definition, however, was inadequate for the precision requirements of modern technology, and the metric kilogram was redefined in terms of a manmade physical object.

Gravitational Mass

Active Gravitational mass is a property of the mass of an object that produces a gravitational field in the space surrounding the object. Presumably, the gravitational field created by one object interacts with the passive gravitational masses of other objects to produce gravitational forces, and these gravitation forces govern large-scale structures in the universe. Gravitational forces hold the galaxies together. They cause clouds of gas and dust to coalesce into stars and planets. They provide the necessary pressure for nuclear fusion to occur within stars. And they determine the orbits of various objects within the solar system. Since gravitational effects are all around us, it is impossible to pin down the exact date when humans first discovered gravitational mass. However, it is possible to identify some of the significant steps towards our modern understanding of gravitational mass and its relationship to the other mass phenomena.

The Astrological Universe

Zodiac in a 6th century synagogue at Beit Alpha, Israel.

The phrase action at a distance was used with early theories of gravitation to describe how an object could be affected by another object which was separated in space and with no known mediator of the interaction. Although this is a relatively new idea, the underlying concept that heavenly objects and motions mysteriously influence circumstances and events on earth is very old. The justification for this belief is not hard to find since heavenly motions clearly do govern the passage of day and night and yearly seasonal changes, and they also govern the daily rise and fall of the tides. Humans from many ancient cultures appear to have been avid stargazers, and part of their motivation was to synchronize heavenly motions with earthly events. The western zodiac, as depicted in the image to the right, represents the apparent yearly path of the Sun across the heavens. In the western zodiac, the ecliptic is divided into twelve equal zones of celestial longitude.

Stone carving of Chinese zodiac

A stone carving of the Chinese zodiac is depicted in the image to the left. Chinese astronomers built this system (know as the earthly branches) from observations of the orbit of Jupiter (歳星 Suìxīng, the Year Star), which has an 11.86 yr period. Chinese astronomers divided the celestial circle into 12 sections to follow the orbit of Jupiter, and assigned an animal to each year. These earthly branches were cyclically paired with celestial stems, a base ten numeral system, to produce a 60 year sexagenary cycle, and each year was assigned a Tai Sui deity to be worshipped, or at least respected during that year.

The sun stone also called the Aztec calendar on display at the National Museum of Anthropology in Mexico City.

A stone carving of the Aztec calendar is depicted in the image to the right. The astronomical systems used by early Americans have surprising similarities with some Asian systems. The Asians obtained their 60 year sexagenary cycle by cyclically pairing the base ten celestial stems with the base twelve earthly branches, the least common multiple of 10 and 12 being 60. The Americans obtained a 260 day tonalpohualli (Mayan Tzolkin) cycle by pairing their base twenty numeral system with a base thirteen trecena cycle, the least common multiple of 20 and 13 being 260.

The word planet comes from the Greek verb πλανώμαι planōmai which means to wander around. The planets appear to move through the night sky and are thus distinguished from the stars, which appear to maintain a fixed position with respect to each other. This supposed ability to move freely may have given the planets the appearance of self-determination. Consequently, many cultures have either directly worshipped the planets as deities or at least associated them with divinity. In fact, the modern English names for the planets were derived from their names as Roman gods.

The Monotheistic Universe

God the Geometer: illustration from a 13th century manuscript

The rise of Judeo-Christian-Islamic monotheism in the west saw a precipitous decline in planet worship, which was explicitly forbidden in many monotheistic faiths. The monotheistic faiths; however, did allow or even encourage the idea that heavenly motions were an expression of God’s perfection. The 13th century illustration to the left depicts God using a compass in the divine act of creation. In the monotheistic world view, the planets were not divine entities moving about in a self-deterministic fashion, but rather, they were God’s creations moving according to a divinely ordained schedule. And understanding that schedule was a means for humans to better understand their creator.

Kepler's Platonic solid model of the Solar system from Mysterium Cosmographicum (1600)

Johannes Kepler was the first person to give an accurate description of the orbits of the planets, and by doing so; he was also the first person to describe active gravitational mass. Kepler lived in an era when there was no clear distinction between astronomy and astrology, and when these fields of study, together with geometry, were viewed as intrinsically divine. Kepler was motivated by religious convictions and incorporated religious arguments and reasoning in his work.

Kepler’s first attempt to describe planetary orbits was similar to the Wu Xing philosophy employed previously by the Chinese. The Chinese had reasoned that there are five directions on a compass, five stages in a process (each with an associated element), and five planets visible to the naked eye. The Chinese therefore associated each planet with a particular direction and element. In a similar fashion, Greek philosophers approximately two thousands years prior to Kepler had proven that there were exactly five Platonic solids. The Greek philosophers had reasoned that each solid must be associated with a specific element, and Kepler further reasoned that perhaps the distances between the planetary orbits could be determined by placing Platonic solids inside of concentric spheres. This theory was somewhat successful, but didn’t agree with available astronomical data to the level of precision that Kepler desired.

Keplerian Gravitational Mass

After the failure of Kepler’s Platonic solids model, he began experimenting with an array of traditional astronomical methods. In 1600 AD, Kepler sought employment with Tycho Brahe and consequently gained access to astronomical data of a higher precision than any previously available. Using Brahe’s precise observations of the planet Mars, Kepler was able to prove that none of the traditional astronomical methods accurately predicted the orbit of Mars, and Kepler spent the next five years developing his own method for determining planetary orbits.

English Name		The Divine Planets					The Keplerian Planets
		Babylonian deity	Greek deity	Hindu Navagraha	Chinese Wu Xing		Semi-major axis	Sidereal orbital period	Mass of Sun
Mercury		Nabu	Hermes	Budha	Black Tortoise		0.387 099 AU	0.240 842 sidereal year
Venus		Ishtar	Aphrodite	Shukra	White Tiger		0.723 332 AU	0.615 187 sidereal year
Mars		Nergal	Ares	Mangala	Vermilion Bird		1.523 662 AU	1.880 816 sidereal year
Jupiter		Marduk	Zeus	Brihaspati	Azure Dragon		5.203 363 AU	11.861 776 sidereal year
Saturn		Ninurta	Cronus	Shani	Yellow Dragon		9.537 070 AU	29.456 626 sidereal year

Kepler’s final successful planetary model in some ways reflected Kepler’s own religious convictions. Kepler reasoned that the sun was representative of the monotheistic God of Christianity, that the sun sat in the center of the solar system and controlled the motions of all other objects in the solar system. Kepler further reasoned that since the sun was the source of motion, then an object’s motion should be inversely proportional to its distance from the sun^[3][4]. In other words, the closer an object gets to the sun the faster it moves (Kepler later refined this to state that an orbit sweeps out equal areas in equal times). Kepler also determined that planets follow elliptical paths as they orbit the sun; and finally, that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. These three rules are known as Kepler's laws of planetary motion.

Illustration of Kepler's three laws with two planetary orbits. (1) The orbits are ellipses with the sun located in a focal point. (2) The two shaded sectors A₁ and A₂ have the same surface area and the time to cover segment A₁ is equal to the time to cover segment A₂. (3) The total orbit times for planet 1 and planet 2 have a ratio a₁^3/2 : a₂^3/2.

The concept of active gravitational mass is an immediate consequence of Kepler’s laws. Prior to Kepler’s time, the planets were thought to move on their own without being compelled by an outside force. But Kepler described how the planet's motions are a direct result of the sun acting upon them. Kepler’s laws also provide a means for quantifying gravitational mass. Since the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis, then the ratio of these two values must be constant for all planets in the solar system. This constant ratio is a direct measure of the sun's active gravitational mass, it has units of distance cubed per time squared, and it is known as the standard gravitational parameter.

A comparison of Kepler’s elliptical planetary model with earlier concepts of the solar system illustrates the importance of having complete and accurate observational data. Ancient astronomers from many civilizations had carefully observed the apparent motions of the sun, moon, and planets against the fixed background stars; and from these observations, they had determined the approximate times required for each of the heavenly bodies to complete a cycle. The ancient astronomers; however, were unable to determine the exact paths followed by these heavenly objects and the distances involved in completing these cycles. Therefore, given this incomplete state of knowledge (knowing the orbital periods but not the paths and distances), the ancient astronomers were unable to accurately describe the solar system, and hence, were unable to understand the gravitational effects of the sun.

In 1609, Johannes Kepler published a careful analysis of the orbit of Mars, explaining how Mars follows an elliptical orbit under the influence of the sun. On August 25 of that same year, Galileo Galilei demonstrated his first telescope to a group of Venetian merchants. Galileo has been called the "father of modern observational astronomy,"^[5] the "father of modern physics,"^[6] and "the Father of Modern Science."^[7] These claims are obviously subjective, and have been disputed, but clearly Galileo was an important figure in the development of scientific instrumentation.

English Name		The Galilean moons
		Semi-major axis	Sidereal orbital period	Mass of Jupiter
Io		0.002 819 AU	0.004 843 sidereal year
Europa		0.004 486 AU	0.009 722 sidereal year
Ganymede		0.007 155 AU	0.019 589 sidereal year
Callisto		0.012 585 AU	0.045 694 sidereal year

Galileo was not the inventor of the telescope, nor was he the first person to build one, yet Galileo’s use of the telescope for scientific research was so revolutionary and groundbreaking that his name has ever been associated with the device. In early January of 1610, Galileo observed four dim objects near Jupiter, which he mistook for stars. However, after a few days of observation, Galileo realized that these objects were in fact orbiting Jupiter. These four objects (later named the Galilean moons in honor of their discoverer) were the first celestial objects observed to orbit something other than the earth or sun. Galileo continued to observe these moons over the next eighteen months, and by mid 1611 he had obtained remarkably accurate estimates for their periods. Later, the semi-major axis of each moon was also estimated, thus allowing the gravitational mass of Jupiter to be determined from the orbits of its moons. The gravitational mass of Jupiter was found to be approximately a thousandth of the gravitational mass of the sun.

Galilean Gravity

Galileo’s use of scientific instrumentation went beyond astronomical observations and the telescope. Sometime prior to 1638, Galileo had turned his attention to the phenomenon of objects falling under the influence of earth’s gravity, and he was actively attempting to characterize these motions. At the time, the relationships between gravity on earth (which caused objects to fall), and gravitational forces in heaven (which caused the planets to orbit) were unknown.

The colors in this image represent variations in Earth’s gravity field as measured by NASA's Gravity Recovery and Climate Experiment (GRACE)

Ball falling freely under gravity. See text for description.

One issue that had puzzled scientists and philosophers was the relationship between an object’s weight and its free fall velocity. The Greek philosopher, Aristotle, had reasoned that every object and every substance has a natural place in the universe, and every object will try to achieve that position. For example, a rock dropped from a building will fall to the earth because the rock is part of the earth and belongs with the other rocks and dirt. Air bubbles in water, on the other hand, will rise because they are composed of air and belong in the atmosphere above the water. Aristotle had also reasoned that free falling objects will fall at a constant speed, and the speed will be inversely proportional to the density of the medium.

Galileo was not the first to investigate earth’s gravitational field, nor was he the first to accurately describe its fundamental characteristics. However, Galileo’s reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. Galileo used a number of scientific experiments to characterize free fall motion. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo ^[8], but the results obtained from these experiments were both realistic and compelling. A biography by Galileo's pupil Vincenzo Viviani stated that Galileo had dropped balls of the same material, but different masses, from the Leaning Tower of Pisa to demonstrate that their time of descent was independent of their mass.^[9]

A second experiment was described in Galileo’s Two New Sciences published in 1638. One of Galileo’s fictional characters, Salviati, describes an experiment using a bronze ball and a wooden ramp. The wooden ramp was "12 cubits long, half a cubit wide and three finger-breadths thick" with a straight, smooth, polished groove. The groove was lined with "parchment, also smooth and polished as possible". And into this groove was placed "a hard, smooth and very round bronze ball". The ramp was inclined at various angles to slow the acceleration enough so that the elapsed time could be measured. The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured. The time was measured using a water clock described as follows:

"a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.".^[10]

Inertial and gravitational mass

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In classical mechanics, Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. That it does is merely an empirical fact.

Albert Einstein developed his general theory of relativity starting from the assumption that this correspondence between inertial and (passive) gravitational mass is not accidental: that no experiment will ever detect a difference between them (the weak version of the equivalence principle). However, in the resulting theory gravitation is not a force and thus not subject to Newton's third law, so "the equality of inertial and active gravitational mass [...] remains as puzzling as ever".^[11]

Inertial mass

This section uses mathematical equations involving differential calculus.

Inertial mass is the mass of an object measured by its resistance to acceleration.

To understand what the inertial mass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Later on, we will see how our classical definition of mass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics. However, the implications of special relativity will not change the meaning of "mass" in any essential way.

According to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion

where f is the force acting on the body and v is its velocity. For the moment, we will put aside the question of what "force acting on the body" actually means.

Now, suppose that the mass of the body in question is a constant. This assumption, known as the conservation of mass, rests on the ideas that (i) mass is a measure of the amount of matter contained in a body, and (ii) matter can never be created or destroyed, only split up or recombined. These are very reasonable assumptions for everyday objects, though, as we will see, matter can indeed be created or destroyed if "matter" is defined strictly as certain kinds of particles and not others. However, (see below) in theory of relativity all mathematically definable definitions of mass are separately conserved over time within closed systems (where no particles or energy are allowed into or out of the system), because energy is conserved over time in such systems, and mass and energy in relativity always occur in exact association.

When the mass of a body is constant (neither mass nor energy are being allowed in or out of the body), Newton's second law becomes

where a denotes the acceleration of the body.

This equation illustrates how mass relates to the inertia of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force.

However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of Newton's third law, which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects A and B, with constant inertial masses m_A and m_B. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on A by B, which we denote f_AB, and the force exerted on B by A, which we denote f_BA. As we have seen, Newton's second law states that

and

where a_A and a_B are the accelerations of A and B, respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that

Substituting this into the previous equations, we obtain

Note that our requirement that a_A be non-zero ensures that the fraction is well-defined.

This is, in principle, how we would measure the inertial mass of an object. We choose a "reference" object and define its mass m_B as (say) 1 kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations.

Newtonian Gravitational mass

The Newtonian concept of gravitational mass rests on Newton's law of gravitation. Let us suppose we have two objects A and B, separated by a distance |r_AB|. The law of gravitation states that if A and B have gravitational masses M_A and M_B respectively, then each object exerts a gravitational force on the other, of magnitude

where G is the universal gravitational constant. The above statement may be reformulated in the following way: if g is the acceleration of a reference mass at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is

This is the basis by which masses are determined by weighing. In simple bathroom scales, for example, the force f is proportional to the displacement of the spring beneath the weighing pan (see Hooke's law), and the scales are calibrated to take g into account, allowing the mass M to be read off. Note that a balance (see the subheading within Weighing scale) as used in the laboratory or the health club measures gravitational mass; only the spring scale measures weight.

Equivalence of inertial and gravitational masses

The equivalence of inertial and gravitational masses is sometimes referred to as the Galilean equivalence principle or weak equivalence principle. The most important consequence of this equivalence principle applies to freely falling objects. Suppose we have an object with inertial and gravitational masses m and M respectively. If the only force acting on the object comes from a gravitational field g, combining Newton's second law and the gravitational law yields the acceleration

This says that the ratio of gravitational to inertial mass of any object is equal to some constant K if and only if all objects fall at the same rate in a given gravitational field. This phenomenon is referred to as the 'universality of free-fall'. (In addition, the constant K can be taken to be 1 by defining our units appropriately.)

The first experiments demonstrating the universality of free-fall were conducted by Galileo. It is commonly stated that Galileo obtained his results by dropping objects from the Leaning Tower of Pisa, but this is most likely apocryphal; actually, he performed his experiments with balls rolling down inclined planes. Increasingly precise experiments have been performed, such as those performed by Loránd Eötvös, using the torsion balance pendulum, in 1889. As of 2008^[update], no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the accuracy 10^-12. More precise experimental efforts are still being carried out.

The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especially friction and air resistance, must be absent or at least negligible. For example, if a hammer and a feather are dropped from the same height through the air on Earth, the feather will take much longer to reach the ground; the feather is not really in free-fall because the force of air resistance upwards against the feather is comparable to the downward force of gravity. On the other hand, if the experiment is performed in a vacuum, in which there is no air resistance, the hammer and the feather should hit the ground at exactly the same time (assuming the acceleration of both objects towards each other, and of the ground towards both objects, for its own part, is negligible). This can easily be done in a high school laboratory by dropping the objects in transparent tubes that have the air removed with a vacuum pump. It is even more dramatic when done in an environment that naturally has a vacuum, as David Scott did on the surface of the Moon during Apollo 15.

A stronger version of the equivalence principle, known as the Einstein equivalence principle or the strong equivalence principle, lies at the heart of the general theory of relativity. Einstein's equivalence principle states that within sufficiently small regions of space-time, it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that the force acting on a massive object due to a gravitational field is a result of the object's tendency to move in a straght line (in other words its inertia) and should therefore be a function of its inertial mass and the strength of the gravitational field.

Mass and energy in relativity

The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant mass is another name for the rest mass of single particles. However, the more general invariant mass (calculated with a more complicated formula) may also be applied to systems of particles in relative motion, and because of this, is usually reserved for systems which consist of widely separated high-energy particles. The invariant mass of systems is the same for all observers and inertial frames, and cannot be destroyed, and is thus conserved, so long as the system is closed. In this case, "closure" implies that an idealized boundary is drawn around the system, and no mass/energy is allowed across it. In as much as energy is conserved in closed systems in relativity, the relativistic definition(s) of mass are quantities which are conserved; also, they do not change over time, even as some types of particles are converted to others.

In bound systems, the binding energy must (often) be subtracted from the mass of the unbound system, simply because this energy has mass, and this mass is subtracted from the system when it is given off, at the time it is bound. Mass is not conserved in this process because the system is not closed during the binding process. A familiar example is the binding energy of atomic nuclei, which appears as other types of energy (such as gamma rays) when the nuclei are formed, and (after being given off) results in nuclides which have less mass than the free particles (nucleons) of which they are composed.

The term relativistic mass is also used, and this is the total quantity of energy in a body or system (divided by c²). The relativistic mass (of a body or system of bodies) includes a contribution from the kinetic energy of the body, and is larger the faster the body moves, so unlike the invariant mass, the relativistic mass depends on the observer's frame of reference. However, for given single frames of reference and for closed systems, the relativistic mass is also a conserved quantity.

Because the relativistic mass is proportional to the energy, it has gradually fallen into disuse among physicists.^[12] There is disagreement over whether the concept remains pedagogically useful.^[13][14]

For a discussion of mass in general relativity, see mass in general relativity.

Notes

References

• R.V. Eötvös et al., Ann. Phys. (Leipzig) 68 11 (1922)
• E.F. Taylor, J.A. Wheeler (1992). Spacetime Physics. New York: W.H. Freeman. ISBN 0-7167-2327-1. 

External links

• Stanford Encyclopedia of Philosophy: "The Equivalence of Mass and Energy" by Francisco Flores.
• "The Mysteries of Mass," Scientific American, July 2005.
• Usenet Physics FAQ by John Baez:
  • "Does mass change with velocity?"
  • "What is the mass of a photon?"
• "The Origin of Mass and the Feebleness of Gravity." Video of lecture by the Nobel Laureate Frank Wilczek.
• Okun, L. B., "Photons, Clocks, Gravity and the Concept of Mass." Slides for a talk.
• The Apollo 15 Hammer-Feather Drop.
• Online mass units conversion.
• Jerome, Jacky (2009) "Mass and Gravity."