No differences.
Additions:
"" The twin paradox considers a situation precisely analogous to the change in the rate of clocks due to a gravitational field. One twin travels on a spacecraft at high velocity, and then returns, while the other remains on earth. On both the outward and return journeys, the spacecraft clock runs slow compared to the Earth clock, so the returning twin ages less than the one who stays at home. But from the point of view of the traveller, the Earth clock was running slow, so why does the twin who stays at home not age less?
The resolution is that the situation is not symmetrical. The travelling twin is subject to active forces at the point when he turns around to come back. As a result, in coordinates defined from spacecraft using the radar method, the Earth clock goes fast during the period when the radar signal is emitted on the outward journey and returns on the homeward journey. |
""
Deletions:
""| The twin paradox considers a situation precisely analogous to the change in the rate of clocks due to a gravitational field. One twin travels on a spacecraft at high velocity, and then returns, while the other remains on earth. On both the outward and return journeys, the spacecraft clock runs slow compared to the Earth clock, so the returning twin ages less than the one who stays at home. But from the point of view of the traveller, the Earth clock was running slow, so why does the twin who stays at home not age less? |
""
""| The resolution is that the situation is not symmetrical. The travelling twin is subject to active forces at the point when he turns around to come back. As a result, in coordinates defined from spacecraft using the radar method, the Earth clock goes fast during the period when the radar signal is emitted on the outward journey and returns on the homeward journey. |
""
Additions:
Determining when reference matter is inertial presents a practical problem. The forces on an object are determined through changes in motion using Newton’s second law, but reference matter cannot show changes motion with respect to itself, even in principle. It does not help to determine motion of reference matter with respect to other matter. That only shifts the problem and leads to circular argument.
Deletions:
There is a practical problem of determining when reference matter is inertial. The forces on an object are determined through changes in motion using Newton’s second law, but reference matter cannot show changes motion with respect to itself, even in principle. It does not help to determine motion of reference matter with respect to other matter. That only shifts the problem and leads to circular argument.
No differences.
Additions:
>>""Note: Local means local in time as well as in space."" >>Einstein’s solution has three parts. First, uniform motion with respect to a background is replaced by uniform motion with respect to other inertial matter. Second, according to the ""general principle of relativity"", laws of physics hold locally, not globally. The motions of satellite and of the blue and green object are close to linear over sections of the orbit. The first law is thus restated:
This makes it possible to define an inertial reference frame:
Finally, the laws of physics are expressed in a form which applies in all reference frames, no matter what active forces may be present on reference matter used to define coordinates. We will see how to do this using the ""principle of general covariance"", //The laws of physics have tensorial form//. Using general covariance, we could replace the 3-vector quantities in N2 and N3 with replaced by 4-vectors in spacetime, but the 4-vector quantity known as [[http://en.wikipedia.org/wiki/Proper_acceleration proper acceleration]], does not correspond directly to 3-vector acceleration and the concept of force is of little use in general relativity, because conservation of energy-momentum has a deeper role and is easier to apply. There is a better way forward.
The consequence of the second and third laws taken together is that the active and reactive forces produce equal and opposite changes in momentum in the bodies they act on. Likewise one can see, from N2 and N3, that, in an inertial reference frame, equal and opposite changes in momentum must be produced by equal and opposite forces. Then second and third laws can be replaced by an equivalent law, //conservation of momentum//.
Conservation of momentum is incorporated (as conservation of the 4-vector quantity, energy-momentum) as a fundamental assumption of general relativity, in Einstein’s field equation. One should therefore not say that general relativity shows Newtonian mechanics wrong, but rather that it incorporates Newtonian mechanics, and indeed reduces to Newtonian mechanics in situations where there is no practical difference between an inertial reference frame and Newton’s concepts of absolute space and time.
I don’t like the description of inertial forces as fictitious because we actually experience them. We experience [[http://en.wikipedia.org/wiki/G-force g-forces]] in a car when it accelerates, slows down, or goes round a corner. On a playground roundabout, or carousel, the [[http://en.wikipedia.org/wiki/Centrifugal_force centrifugal force]] pulls you outward from the centre, and the [[http://en.wikipedia.org/wiki/Coriolis_force Coriolis force]] pulls you sideways when you move.
""
""The non-existence of the centrifugal force is often drummed into school children. Relativistically, this is completely wrong. The centrifugal force does not exist within Newton’s formulation in terms of absolute space, but if there is no absolute space it becomes impossible to exclude it from physical law. From the perspective of an external observer, there is no centrifugal force; the boy is being accelerated toward the teacher and if the rope were to break he would fly on in a constant direction of motion. That is one description. From boy’s perspective, with respect to the reference matter of his own body, the centrifugal force is very real. If any schoolteachers wish to teach otherwise, let them go to a children’s playground and play on the roundabout until they convince themselves of the fact. It only makes sense to talk of motion, of acceleration, or of force, relative to chosen reference matter.
>>""Definition: A Newton is the amount of net force required to accelerate a mass of one kilogram at a rate of one meter per second per second.""
>>The replacement of absolute space with a local reference frame introduces inertial forces as an intrinsic part of the definition of 4-vectors, but otherwise, Newton’s second law remains unchanged. N2 is actually the definition of a force in Newtonian mechanics, and is used in the definition of the [[http://en.wikipedia.org/wiki/Newton_(unit) Newton]], the internationally agreed the unit of force. According to ""N2"", //a force is anything which causes an acceleration within a given reference frame//. This includes inertial forces resulting from the choice of coordinates. Einstein realised that the force of gravity could also be understood as an inertial force, and encapsulated this insight in the equivalence principle.
{{image class="right" alt="EquivalencePrinciple-4" title="Einstein’s Lift Experiment" url="images/equivalenceprinciple/EquivalencePrinciple-4N.gif"}}{{image class="left" alt="EquivalencePrinciple-5" title="Weightlessness training" url="images/equivalenceprinciple/EquivalencePrinciple-5N.gif"}}A clear distinction between active and inertial forces is that an active force is applied to a part of a body, and is transmitted through the body by means of further active forces in the structure of matter, whereas inertial forces apply equally on every part of a body, are not transmitted through it, and are directly proportional to mass. This is how we experience the force of gravity. Einstein began to think that the force of gravity might be an inertial force. In his lift experiment, if the chord were to break and the lift were to go into free fall, a man in the lift could find no experiment to determine either the existence of a gravitational field or the motion of the lift without looking outside the lift. The principle in weightlessness training for astronauts is the same; an aircraft flies on a parabolic path equivalent to that of free fall in the absence of atmospheric resistance. As a result, the occupants of the aircraft experience no gravitational field, exactly as they would in space. Free fall theme park rides like [[http://www.altontowers.com/content.php?areaid=1&pageid=17 Oblivion]] also provide an experience of weightlessness (at least, if you shut your eyes).
"" Einstein considered the situation where a constant active force is applied continuously, on a reference frame, causing constant acceleration of the frame with respect to inertial matter. Because velocity is not constant, lines of equal time are not parallel. Einstein concluded that this must have an effect on the rate of stationary clocks depending on their position within the frame.
Einstein recognised that this is the situation we perceive from the surface of the Earth, which presses up upon objects standing on it, while there are negligible active forces on objects in free fall. The surface of the earth is in constant acceleration relative to objects in free fall. He concluded that a gravitational field affects the rate of clocks, and therefore that it affects other geometrical properties, leading to curvature. |
""
"" The change in frequency of radiation in a gravitational field is known as gravitational redshift. It can also be understood as a loss of energy due to a gravitational potential. If a change in the rate of clocks were not required from the equivalence principle and special relativity, we could also predict it from the change in energy of radiation due to motion in a classical gravitational field. In a spacetime diagram, the start and end of one wavelength are both shown at 45º. Since the wavelength is increasing as the signal rises, the rate of clocks must be reducing.
In this way it is seen that the force of gravity is not merely equivalent to the acceleration of inertial matter due to the geometry of spacetime, but should actually be identified with it. In the absence of an empirical meaning for geometry other than as a set of relationships found through physical measurement processes, any other interpretation is mere metaphysical speculation. Because clocks run faster higher in the gravitational field of the earth, a clock thrown upwards and returning to Earth will measure a greater time than one which remains stationary at the surface of the earth. |
""
""| The twin paradox considers a situation precisely analogous to the change in the rate of clocks due to a gravitational field. One twin travels on a spacecraft at high velocity, and then returns, while the other remains on earth. On both the outward and return journeys, the spacecraft clock runs slow compared to the Earth clock, so the returning twin ages less than the one who stays at home. But from the point of view of the traveller, the Earth clock was running slow, so why does the twin who stays at home not age less? |
""Deletions:
Einstein’s solution has three parts. First, uniform motion with respect to a background is replaced by uniform motion with respect to other inertial matter. Second, according to the ""general principle of relativity"", laws of physics hold locally, not globally. The motions of satellite and of the blue and green object are close to linear over sections of the orbit.
<<""Note: Local means local in time as well as in space.""
The first law is thus restated:
Third, the laws of physics are expressed in a form which applies in all reference frames, no matter what active forces may be present on reference matter used to define coordinates. We will see how to do this using the ""principle of general covariance"". After applying general covariance, 3-vector quantities in Newton’s second law are replaced by 4-vectors in spacetime. At that point we will also know that the change to an accelerating or rotating frame of reference is a ""coordinate transformation"", and that a coordinate transformation is simply a ""change of basis"". Since one basis is precisely equivalent to another, inertial forces associated with accelerating coordinate systems are automatically incorporated in the 4-vector notion of force.
The proof of the existence of inertial forces is that we experience them. We experience [[http://en.wikipedia.org/wiki/G-force g-forces]] in a car when it accelerates, slows down, or goes round a corner. On a playground roundabout, or carousel, the [[http://en.wikipedia.org/wiki/Centrifugal_force centrifugal force]] pulls you outward from the centre, and the [[http://en.wikipedia.org/wiki/Coriolis_force Coriolis force]] pulls you sideways when you move.
""""The non-existence of the centrifugal force is often drummed into school children. In general relativity, this is completely wrong. The centrifugal force does not exist within Newton’s formulation in terms of absolute space, but if there is no absolute space it becomes impossible to exclude it from physical law. From the perspective of an external observer, there is no centrifugal force; the boy is being accelerated toward the teacher and if the rope were to break he would fly on in a constant direction of motion. That is one description. From boy’s perspective, with respect to the reference matter of his own body, the centrifugal force is very real. If any schoolteachers wish to teach otherwise, let them go to a children’s playground and play on the roundabout until they convince themselves of the fact. It only makes sense to talk of motion, of acceleration, or of force, relative to chosen reference matter.
The replacement of absolute space with a local reference frame introduces inertial forces as an intrinsic part of the definition of 4-vectors, but otherwise, Newton’s second law remains unchanged. According to ""N2"", a force is anything which causes an acceleration within a given reference frame. This includes inertial forces resulting from the choice of coordinates.
It is implicit that a reference frame is defined locally, i.e. from local reference matter. The choice of an inertial reference frame is a choice of coordinates, that is a choice of a particular basis. Using this basis, inertial forces do not appear. We may intuitively feel that this choice has a fundamental physical meaning, that it more accurately describes underlying physics in terms of the interactions of matter. However, in the language of tensors used in general relativity, no such distinction is possible. In the mathematical structure of vector space, one basis is equivalent to another. Of course, this does not imply that there is no physical distinction, merely that a distinction cannot be described through the use of tensors in differential geometry.
The second law makes no distinction between active and inertial forces. Not so the third law. In the case of an active force, caused by one body acting upon on another, there is always a reactive force. That is not generally the case for inertial forces, like the centrifugal and Coriolis force. As a result the third law only holds in inertial reference frames, and we cannot directly replace it with a covariant law. The consequence of the second and third laws taken together is that the active and reactive forces produce equal and opposite changes in momentum in the bodies they act on. Likewise one can see, from the first and second laws, that, in an inertial reference frame, equal and opposite changes in momentum must be produced by equal and opposite forces. If, as seems reasonable, we assume that particle interactions take place over very small time intervals, then inertial forces have a negligible effect, and the third law can be replaced by an equivalent law, //conservation of momentum//.
Using conservation of momentum, ""N3"" is effectively re-expressed in terms of 3-vector quantities which can be replaced with 4-vectors. One should not say that general relativity falsifies Newtonian mechanics, but rather that it incorporates Newtonian mechanics, and indeed reduces to Newtonian mechanics in situations where there is no practical difference between an inertial reference frame and Newton’s concepts of absolute space and time.
""| One twin travels on a spacecraft at high velocity, and then returns, while the other remains on earth. On both the outward and return journeys, the spacecraft clock runs slow compared to the Earth clock, so the returning twin ages less than the one who stays at home. But from the point of view of the traveller, the Earth clock was running slow, so why does the twin who stays at home not age less, giving a paradox? |
""
""| Einstein considered the situation where a constant active force is applied continuously, and concluded that this must have an effect on the rate of stationary clocks depending on their position. He recognised that this is the situation we percieve from the surface of the Earth, which presses up upon objects standing on it, while there are negligible active forces on objects in free fall. He concluded that a gravitational field affects the rate of clocks, and therefore that it affects other geometrical properties, leading to curvature. |
""
{{image class="right" alt="EquivalencePrinciple-4" title="Einstein’s Lift Experiment" url="images/equivalenceprinciple/EquivalencePrinciple-4N.gif"}}{{image class="left" alt="EquivalencePrinciple-5" title="Weightlessness training" url="images/equivalenceprinciple/EquivalencePrinciple-5N.gif"}}A clear distinction between active and inertial forces is that an active force is applied to a part of a body, and is transmitted through the body by means of further active forces in the structure of matter, whereas inertial forces apply equally on every part of a body, are not transmitted through it, and are directly proportional to mass. This is how we experience the force of gravity. Einstein began to think that the force of gravity might be an inertial force. In his lift experiment, if the chord were to break and the lift were to go into free fall, a man in the lift could find no experiment to determine either the existence of a gravitational field or the motion of the lift without looking outside the lift. The principle in weightlessness training for astronauts is the same; an aircraft flies on a parabolic path equivalent to that of free fall in the absence of atmospheric resistance. As a result, the occupants of the aircraft experience no gravitational field, exactly as they would in space. Free fall theme park rides like [[http://www.altontowers.com/content.php?areaid=1&pageid=17 Oblivion]] also provide an experience of weightlessness.
The change in frequency of radiation in a gravitational field is known as [[http://en.wikipedia.org/wiki/Gravitational_redshift gravitational redshift]]. It can also be understood as a loss of energy due to a gravitational potential. If a change in rate of clocks were not required from the equivalence principle and special relativity, we could also predict it from the change in energy of radiation due to motion in a classical gravitational field. Although there are physicists looking for gravitons as intermediaries for the gravitational force, by analogy with photons in quantum electrodynamics, it appears to me that the force gravity is not merely equivalent to the acceleration of inertial matter due to the geometry of spacetime, but should actually be identified with it. In the absence of an empirical meaning for geometry other than as a set of relationships found through physical measurement processes, any other interpretation is mere metaphysical speculation.
Additions:
"" If an active force is present on the reference matter used to define coordinates then we cannot expect an inertial object to remain in uniform motion in those coordinates. According to the stewardess and passenger of a plane, Newton’s laws hold while the the plane is in constant motion, but they are subjected to g-forces during take-off and landing, and a freely falling object would be seen to accelerate with respect to the plane.
There is a practical problem of determining when reference matter is inertial. The forces on an object are determined through changes in motion using Newton’s second law, but reference matter cannot show changes motion with respect to itself, even in principle. It does not help to determine motion of reference matter with respect to other matter. That only shifts the problem and leads to circular argument.
Deletions:
""| If an active force is present on the reference matter used to define coordinates then we cannot expect an inertial object to remain in uniform motion in those coordinates. According to the stewardess and passenger of a plane, Newton’s laws hold while the the plane is in constant motion, but they are subjected to g-forces during take-off and landing, and a freely falling object would be seen to accelerate with respect to the plane. There is a practical problem of determining when reference matter is inertial. The forces on an object are determined through changes in motion using Newton’s second law, but reference matter cannot show changes motion with respect to itself, even in principle. It does not help to determine motion of reference matter with respect to other matter. That only shifts the problem and leads to circular argument. |
Additions:
Deletions:
====""""Newton’s Laws of Motion====
====""""Inertial Matter====
====""""Active and Inertial Forces====
====""""The Twin Paradox====
====""""The Equivalence Principle====
====""""The Pound-Rebka Experiment====
====""""Gravitational Red Shift====
====""""Mach’s Principle====
Additions:
====""""Newton’s Laws of Motion====
====""""Inertial Matter====
====""""Active and Inertial Forces====
====""""The Twin Paradox====
====""""The Equivalence Principle====
====""""The Pound-Rebka Experiment====
====""""Gravitational Red Shift====
====""""Mach’s Principle====
[[TheEquivalencePrinciple Einstein’s Equivalence Principle ↑]] [[MathematicalMethods Mathematical Methods →]]
Deletions:
====""""Newton’s Laws of Motion====
====""""Inertial Matter====
====""""Active and Inertial Forces====
====""""The Twin Paradox====
====""""The Equivalence Principle====
====""""The Pound-Rebka Experiment====
====""""Gravitational Red Shift====
====""""Mach’s Principle====
[[TheEquivalencePrinciple Einstein’s Equivalence Principle ↑]] [[MathematicalMethods Mathematical Methods →]
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