Non-Euclidean geometry and the rotating disk Einstein's formulation of special relativity was in terms of kinematics (the study of moving bodies without reference to forces). Late in 1907, his former mathematics professor, Hermann Minkowski, presented an alternative, geometric interpretation of special relativity in a lecture to the Göttingen Mathematical society, introducing the concept of spacetime. Einstein was initially dismissive of Minkowski's geometric interpretation, regarding it as überflüssige Gelehrsamkeit (superfluous learnedness). 300px|Consideration of the Ehrenfest paradox led Einstein to consider that gravitation curves spacetime. As with special relativity, Einstein's early results in developing what was ultimately to become general relativity were accomplished using kinematic analysis rather than geometric techniques of analysis. Starting with the   equivalence principle, Einstein deduced the phenomena of gravitational time dilation, the gravitational bending of light (albeit with the wrong value )   , and the gravitational slowing of the speed of light .            By 1912, however, Einstein had reached an impasse  , realizing that he needed to go beyond the mathematics that he knew and was familiar with.
Early applications of the equivalence principle Einstein's formulation of special relativity was in terms of kinematics (the study of moving bodies without reference to forces). Late in 1907, his former mathematics professor, Hermann Minkowski, presented an alternative, geometric interpretation of special relativity in a lecture to the Göttingen Mathematical society, introducing the concept of spacetime. Einstein was initially dismissive of Minkowski's geometric interpretation, regarding it as überflüssige Gelehrsamkeit (superfluous learnedness). As with special relativity, Einstein's early results in developing what was ultimately to become general relativity were accomplished using kinematic analysis rather than geometric techniques of analysis. In his 1907 Jahrbuch paper, Einstein first addressed the question of whether the propagation of light is influenced by gravitation, and whether there is any effect of a gravitational field on clocks. In 1911, Einstein returned to this subject, in part because he had realized that certain predictions of his nascent theory were amenable to experimental test. By the time of his 1911 paper, Einstein and other scientists had offered several alternative demonstrations that the inertial mass of a body increases with its energy content: If the energy increase of the body is , then the increase in its inertial mass is Einstein asked whether there is an increase of gravitational mass corresponding to the increase in inertial mass, and if there is such an increase, is the increase in gravitational mass precisely the same as its increase in inertial mass? Using the equivalence principle, Einstein concluded that this must be so. Einstein's argument that falling light acquires energy To show that the equivalence principle necessarily implies the gravitation of energy, Einstein considered a light source separated along the z‑axis by a distance above a receiver in a homogeneous gravitational field having a force per unit mass of 1 A certain amount of electromagnetic energy is emitted by towards According to the equivalence principle, this system is equivalent to a gravitation-free system which moves with uniform acceleration in the direction of the positive z‑axis, with separated by a constant distance from In the accelerated system, light emitted from takes (to a first approximation) to arrive at But in this time , the velocity of will have increased by from its velocity when the light was emitted. The energy arriving at will therefore not be the energy but the greater energy given by According to the equivalence principle, the same relation holds for the non-accelerated system in a gravitational field, where we replace by the gravitational potential difference between and so that The energy arriving at is greater than the energy emitted by by the potential energy of the mass in the gravitational field. Hence corresponds to the gravitational mass as well as the inertial mass of a quantity of energy. Einstein's 1911 thought experiment to demonstrate that the energy of gravitational mass must equal the energy of inertial mass To further clarify that the energy of gravitational mass must equal the energy of inertial mass, Einstein proposed the following cyclic process: (a) A light source is situated a distance above a receiver in a uniform gravitational field. A movable mass can shuttle between and (b) A pulse of electromagnetic energy is sent from to The energy is absorbed by (c) Mass is lowered from to releasing an amount of work equal to (d) The energy absorbed by is transferred to This increases the gravitational mass of to a new value (e) The mass is lifted back to , requiring the input of work (e) The energy carried by the mass is then transferred to completing the cycle. Conservation of energy demands that the difference in work between raising the mass and lowering the mass, , must equal or one could potentially define a perpetual motion machine. Therefore, In other words, the increase in gravitational mass predicted by the above arguments is precisely equal to the increase in inertial mass predicted by special relativity. Einstein then considered sending a continuous electromagnetic beam of frequency (as measured at ) from to in a homogeneous gravitational field. The frequency of the light as measured at will be a larger value given by Einstein noted that the above equation seemed to imply something absurd: Given that the transmission of light from to is continuous, how could the number of periods emitted per second from be different from that received at It is impossible for wave crests to appear on the way down from to . The simple answer is that this question presupposes an absolute nature of time, when in fact there is nothing that compels us to assume that clocks situated at different gravitational potentials must be conceived of as going at the same rate. The principle of equivalence implies gravitational time dilation. It is important to realize that Einstein's arguments predicting gravitational time dilation are valid for any theory of gravity that respects the principle of equivalence. This includes Newtonian gravitation. Experiments such as the Pound–Rebka experiment, which have firmly established gravitational time dilation, therefore do not serve to distinguish general relativity from Newtonian gravitation. In the remainder of Einstein's 1911 paper, he discussed the bending of light rays in a gravitational field, but given the incomplete nature of Einstein's theory as it existed at the time, the value that he predicted was half the value that would later be predicted by the full theory of general relativity. (ArXiv version here: arxiv.org/abs/1403.7377.)Ned Wright: Deflection and Delay of Light Non-Euclidean geometry and the rotating disk 300px|Consideration of the Ehrenfest paradox led Einstein to consider that gravitation curves spacetime. By 1912, Einstein had reached an impasse in his kinematic development of general relativity , realizing that he needed to go beyond the mathematics that he knew and was familiar with.
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Non-Euclidean geometry and the rotating disk
Early applications of the equivalence principle
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300px|Consideration of the Ehrenfest paradox led Einstein to consider that gravitation curves spacetime.
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Starting with the
In his 1907 Jahrbuch paper, Einstein first addressed the question of whether the propagation of light is influenced by gravitation, and whether there is any effect of a gravitational field on clocks. In 1911, Einstein returned to this subject, in part because he had realized that certain predictions of his nascent theory were amenable to experimental test.
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By the time of his 1911 paper, Einstein and other scientists had offered several alternative demonstrations that the inertial mass of a body increases with its energy content: If the energy increase of the body is , then the increase in its inertial mass is
A
Einstein asked whether there is an increase of gravitational mass corresponding to the increase in inertial mass, and if there is such an increase, is the increase in gravitational mass precisely the same as its increase in inertial mass? Using the
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deduced the phenomena of gravitational time dilation, the gravitational bending of light (albeit with the wrong value )
concluded that this must be so.
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Einstein's argument that falling light acquires energy To show that the equivalence principle necessarily implies the gravitation of energy, Einstein considered a light source separated along the z‑axis by a distance above a receiver in a homogeneous gravitational field having a force per unit mass of 1 A certain amount of electromagnetic energy is emitted by towards According to the equivalence principle, this system is equivalent to a gravitation-free system which moves with uniform acceleration in the direction of the positive z‑axis, with separated by a constant distance from
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In the accelerated system, light emitted from takes (to a first approximation) to arrive at But in this time
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and
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gravitational slowing of the speed of light .
velocity of will have increased by from its velocity when the light was emitted. The energy arriving at will therefore not be the energy but the greater energy given by
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According to the equivalence principle, the same relation holds for the non-accelerated system in a gravitational field, where we replace by the gravitational potential difference between and so that
A
The energy arriving at is greater than the energy emitted by by the potential energy of the mass in the gravitational field. Hence corresponds to the gravitational mass as well as the inertial mass of a quantity of energy.
A
Einstein's 1911 thought experiment to demonstrate that the energy of gravitational mass must equal the energy of inertial mass To further clarify that the energy of gravitational mass must equal the energy of inertial mass, Einstein proposed the following cyclic process: (a) A light source is situated a distance above a receiver in a uniform gravitational field. A movable mass can shuttle between and (b) A pulse of electromagnetic energy is sent from to The energy is absorbed by (c) Mass is lowered from to releasing an amount of work equal to (d) The energy absorbed by is transferred to This increases the gravitational mass of to a new value (e) The mass is lifted back to , requiring the input of work (e) The energy carried by the mass is then transferred to completing the cycle.
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Conservation of energy demands that the difference in work between raising the mass and lowering the mass, , must equal or one could potentially define a perpetual motion machine. Therefore,
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In other words, the increase in gravitational mass predicted by the above arguments is precisely equal to the increase in inertial mass predicted by special relativity.
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Einstein then considered sending a continuous electromagnetic beam of frequency (as measured at ) from to in a homogeneous gravitational field. The frequency of the light as measured at will be a larger value given by
A
Einstein noted that the above equation seemed to imply something absurd: Given that the transmission of light from to is continuous, how could the number of periods emitted per second from be different from that received at It is impossible for wave crests to appear on the way down from to . The simple answer is that this question presupposes an absolute nature of time, when in fact there is nothing that compels us to assume that clocks situated at different gravitational potentials must be conceived of as going at the same rate. The principle of equivalence implies gravitational time dilation.
A
It is important to realize that Einstein's arguments predicting gravitational time dilation are valid for any theory of gravity that respects the principle of equivalence. This includes Newtonian gravitation. Experiments such as the Pound–Rebka experiment, which have firmly established gravitational time dilation, therefore do not serve to distinguish general relativity from Newtonian gravitation.
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In the remainder of Einstein's 1911 paper, he discussed the bending of light rays in a gravitational field, but given the incomplete nature of Einstein's theory as it existed at the time, the value that he predicted was half the value that would later be predicted by the full theory of general relativity. (ArXiv version here: arxiv.org/abs/1403.7377.)Ned Wright: Deflection and Delay of Light
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Non-Euclidean geometry and the rotating disk
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300px|Consideration of the Ehrenfest paradox led Einstein to consider that gravitation curves spacetime.
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however,
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in his kinematic development of general relativity