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Authors: Lawrence M. Krauss
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time as being separate dimensional quantities? We could choose instead to equate the dimensions of length and time. In this case all velocities, which previously had the dimensions of length/time, would now be dimensionless, since the dimensions of length and time in the numerator and denominator would cancel. Physically this is equivalent to writing all velocities as a (dimensionless) fraction of the speed of light, so that if I said that something had a velocity of [1/2], this would mean that its velocity was [1/2] the speed of light. Clearly, this kind of system requires the speed of light to be a universal constant for all observers, so that we can use it as a reference value.
Now we have only two independent dimensional quantities, time and mass (or, equivalently, length and mass). One of the consequences of this unusual system is that it allows us to equate other dimensional quantities besides length and time. For example, Einstein’s famous formula E = mc 2 equates the mass of an object to an equivalent amount of energy. In our new system of units, however, c (= 1) is dimensionless, so that we find that the “dimensions” of energy and mass are now equal. This carries out in practice what Einstein’s formula does formally: It makes a one-to-one connection between mass and energy. Einstein’s formula tells us that since mass can be turned into energy, we can refer to the mass of something in either the units it had before it was transformed into energy or the units of the equivalent amount of energy that it transforms into. We need no longer speak of the
mass of an object in kilograms, or tons, or pounds, but can speak of it in the equivalent units of energy, in, say, “Volts” or “Calories.” This is exactly what elementary-particle physicists do when they refer to the mass of the electron as 0.5 million electron Volts (an electron Volt is the energy an electron in a wire gets when powered by a 1-Volt battery) instead of 10 –31 grams. Since particle-physics experiments deal regularly with processes in which the rest mass of particles is converted into energy, it is ultimately sensible to use energy units to keep track of mass. And that is one of the guidelines: Always use the units that make the most physical sense. Similarly, particles in large accelerators travel at close to the speed of light, so that setting c = 1 is numerically practical. This would not be practical, however, for describing motions on a more familiar scale, where we would have to describe velocities by very small numbers. For example, the speed of a jet airplane in these units would be about 0.000001, or 10 –6 .
Things don’t stop here. There is another universal constant in nature, labeled h and called Planck’s constant, after the German physicist Max Planck (one of the fathers of quantum mechanics). It relates quantities with the dimensions of mass (or energy) to those with the dimensions of length (or time). Continuing as before, we can invent a system of units where not only c = 1 but h = 1. In this case, the relation between dimensions is only slightly more complicated: One finds that the dimension of mass (or energy) becomes equivalent to 1/length, or 1/time. (Specifically, the energy quantity 1 electron Volt becomes equivalent to 1/6 × 10 –16 seconds.) The net result of all this is that we can reduce the three a priori independent dimensional quantities in nature to a single quantity. We can then describe all measurements in the physical world in terms of just one-dimensional quantity, which we can choose to be mass, time, or length at our convenience. To
convert between them, we just keep track of the conversion factors that took us from our normal system of units—in which, for example, the speed of light c = 3 × 10 8 meters/sec—to the system in which c = 1. For example, volume, with the dimensions of length × length × length = length 3 in our normal system of units, equivalently has the dimensions of 1/mass 3
Now we have only two independent dimensional quantities, time and mass (or, equivalently, length and mass). One of the consequences of this unusual system is that it allows us to equate other dimensional quantities besides length and time. For example, Einstein’s famous formula E = mc 2 equates the mass of an object to an equivalent amount of energy. In our new system of units, however, c (= 1) is dimensionless, so that we find that the “dimensions” of energy and mass are now equal. This carries out in practice what Einstein’s formula does formally: It makes a one-to-one connection between mass and energy. Einstein’s formula tells us that since mass can be turned into energy, we can refer to the mass of something in either the units it had before it was transformed into energy or the units of the equivalent amount of energy that it transforms into. We need no longer speak of the
mass of an object in kilograms, or tons, or pounds, but can speak of it in the equivalent units of energy, in, say, “Volts” or “Calories.” This is exactly what elementary-particle physicists do when they refer to the mass of the electron as 0.5 million electron Volts (an electron Volt is the energy an electron in a wire gets when powered by a 1-Volt battery) instead of 10 –31 grams. Since particle-physics experiments deal regularly with processes in which the rest mass of particles is converted into energy, it is ultimately sensible to use energy units to keep track of mass. And that is one of the guidelines: Always use the units that make the most physical sense. Similarly, particles in large accelerators travel at close to the speed of light, so that setting c = 1 is numerically practical. This would not be practical, however, for describing motions on a more familiar scale, where we would have to describe velocities by very small numbers. For example, the speed of a jet airplane in these units would be about 0.000001, or 10 –6 .
Things don’t stop here. There is another universal constant in nature, labeled h and called Planck’s constant, after the German physicist Max Planck (one of the fathers of quantum mechanics). It relates quantities with the dimensions of mass (or energy) to those with the dimensions of length (or time). Continuing as before, we can invent a system of units where not only c = 1 but h = 1. In this case, the relation between dimensions is only slightly more complicated: One finds that the dimension of mass (or energy) becomes equivalent to 1/length, or 1/time. (Specifically, the energy quantity 1 electron Volt becomes equivalent to 1/6 × 10 –16 seconds.) The net result of all this is that we can reduce the three a priori independent dimensional quantities in nature to a single quantity. We can then describe all measurements in the physical world in terms of just one-dimensional quantity, which we can choose to be mass, time, or length at our convenience. To
convert between them, we just keep track of the conversion factors that took us from our normal system of units—in which, for example, the speed of light c = 3 × 10 8 meters/sec—to the system in which c = 1. For example, volume, with the dimensions of length × length × length = length 3 in our normal system of units, equivalently has the dimensions of 1/mass 3
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