gravitational force is proportional to the
square
of the distance between objects (in this case the center of the earth and falling objects at its surface). Doubling the radius without increasing the mass causes a fourfold reduction of gravity.
Some of the more direct ways of measuring this change in gravitational pull would fail. It wouldn’t do to measure the weights of objects in a balance. The balance can only compare the lessened gravitational pull on objects against the equally lessened pull on standard pound or kilogram weights. Schlesinger argued, however, that the weakened gravity could be measured by the height of the mercury column in old-fashioned barometers. The height of the mercury depends on three factors: the air pressure, the density of mercury, and the strength of gravity. Under normal circumstances, only air pressure changes very much.
The air pressure would be eight times less after the doubling, for all volumes would be increased 2 3 times, or eightfold. (You wouldn’t get the bends, though, because your blood pressure would also be eight times less.) The density of mercury would also be eight times less; these two effects would cancel out, leaving the weakened gravity as the measurable change. Since gravity is four times weaker, the mercury should rise four times higher—which will be measured as
two
times higher with the doubled yardsticks. That, then, is a measurable difference.
Schlesinger applied the doubling to some other standard physical laws and further claimed:
• The length of the day, as measured by a pendulum clock, would be 1.414 (the square root of 2) times longer.
• The speed of light would increase by the same factor, as measured by a pendulum clock.
• The year would contain 258 days (365 divided by the square root of 2).
You can quibble with some of Schlesinger’s work. Schlesinger uses a pendulum clock as the unit of time. This clock is much slower because gravity is weaker and the length of the pendulum is doubled. Other types of clocks would not share this slowing. You can argue on the basis of Hooke’s law (which governs the resistance of springs) that an ordinary watch with a mainspring would run at exactly the same rate after the doubling.
There is also the question of whether the usual conservation laws apply during the expansion. Schlesinger supposes that the angular momentum of the earth must remain constant (as it normally would in any possible interaction), even during the doubling. If the earth’s angular momentum is to remain constant, then the rotation of the earth must slow.
There would be other consequences of conservation laws. The universe is mostly hydrogen, which is an electron orbiting a proton. There is electrical attraction between the two particles. To double the size of all atoms is to move all the electrons “uphill,” twice as far away from the protons. This would require a stupendous output of energy. If the law of conservation of mass-energy holds during the doubling, this energy has to come from somewhere. Most plausibly, it would come from a universal lowering of the temperature. Everything would get colder, which would be another consequence of the doubling.
The thrust of Schlesinger’s argument is this: Suppose we got up one morning and found that every mercury barometer in the world had shattered. Investigating further, we find that mercury now rises to about 60 inches rather than 30. (The barometers shattered because no one made the glass columns that long.) Pendulum clocks and spring-wound clocks now keep different time. The velocity of light, when measured by a pendulum clock, is 41.4 percent greater. The length of the year has changed. There are thousands of changes; it is as if all the laws of physics went haywire.
Then someone suggests that what’s happened is that all lengths have doubled. This hypothesis accounts for all the observed changes, and makes many predictions about other changes. Uponhearing of the
square
of the distance between objects (in this case the center of the earth and falling objects at its surface). Doubling the radius without increasing the mass causes a fourfold reduction of gravity.
Some of the more direct ways of measuring this change in gravitational pull would fail. It wouldn’t do to measure the weights of objects in a balance. The balance can only compare the lessened gravitational pull on objects against the equally lessened pull on standard pound or kilogram weights. Schlesinger argued, however, that the weakened gravity could be measured by the height of the mercury column in old-fashioned barometers. The height of the mercury depends on three factors: the air pressure, the density of mercury, and the strength of gravity. Under normal circumstances, only air pressure changes very much.
The air pressure would be eight times less after the doubling, for all volumes would be increased 2 3 times, or eightfold. (You wouldn’t get the bends, though, because your blood pressure would also be eight times less.) The density of mercury would also be eight times less; these two effects would cancel out, leaving the weakened gravity as the measurable change. Since gravity is four times weaker, the mercury should rise four times higher—which will be measured as
two
times higher with the doubled yardsticks. That, then, is a measurable difference.
Schlesinger applied the doubling to some other standard physical laws and further claimed:
• The length of the day, as measured by a pendulum clock, would be 1.414 (the square root of 2) times longer.
• The speed of light would increase by the same factor, as measured by a pendulum clock.
• The year would contain 258 days (365 divided by the square root of 2).
You can quibble with some of Schlesinger’s work. Schlesinger uses a pendulum clock as the unit of time. This clock is much slower because gravity is weaker and the length of the pendulum is doubled. Other types of clocks would not share this slowing. You can argue on the basis of Hooke’s law (which governs the resistance of springs) that an ordinary watch with a mainspring would run at exactly the same rate after the doubling.
There is also the question of whether the usual conservation laws apply during the expansion. Schlesinger supposes that the angular momentum of the earth must remain constant (as it normally would in any possible interaction), even during the doubling. If the earth’s angular momentum is to remain constant, then the rotation of the earth must slow.
There would be other consequences of conservation laws. The universe is mostly hydrogen, which is an electron orbiting a proton. There is electrical attraction between the two particles. To double the size of all atoms is to move all the electrons “uphill,” twice as far away from the protons. This would require a stupendous output of energy. If the law of conservation of mass-energy holds during the doubling, this energy has to come from somewhere. Most plausibly, it would come from a universal lowering of the temperature. Everything would get colder, which would be another consequence of the doubling.
The thrust of Schlesinger’s argument is this: Suppose we got up one morning and found that every mercury barometer in the world had shattered. Investigating further, we find that mercury now rises to about 60 inches rather than 30. (The barometers shattered because no one made the glass columns that long.) Pendulum clocks and spring-wound clocks now keep different time. The velocity of light, when measured by a pendulum clock, is 41.4 percent greater. The length of the year has changed. There are thousands of changes; it is as if all the laws of physics went haywire.
Then someone suggests that what’s happened is that all lengths have doubled. This hypothesis accounts for all the observed changes, and makes many predictions about other changes. Uponhearing of the
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