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Newton’s Laws of Motion

Newton based mechanics on three laws of motion:


In their original form, Newton’s laws describe motions, and changes of motion, with respect to an assumed Euclidean background, absolute space, in conflict with special relativity. General relativity does not dispense with Newton’s laws, but it does dispense with absolute background. As a result, Newton’s laws must be reformulated in terms of motion with respect to coordinates defined from local reference matter. Before we can reformulate the laws, we must analyse the issues which are raised.

Inertial Matter

An inertial object is one which is not subject to active forces. More precisely:

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.

Even when objects do not exert apparent active forces on each other through interaction, they do not, in general, remain indefinitely in a state of uniform motion with respect to each other. Two inertial objects (green and blue) released at rest in a an orbiting satellite are subjected to tidal forces and spontaneously change position with respect to the satellite and with respect to each other.

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:


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.

Active and Inertial Forces

Inertial forces were introduced into physics by Huygens. They are often described as fictitious forces, because they are seen as the result of the acceleration of the coordinate system, not of action by another object. In general relativity, the fiction is that there exists an absolute background with respect to which the acceleration of a coordinate system can be defined.

The proof of the existence of inertial forces is that we experience them. We experience g-forces in a car when it accelerates, slows down, or goes round a corner. On a playground roundabout, or carousel, the centrifugal force pulls you outward from the centre, and the 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.

The Twin Paradox

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?

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.

The Equivalence Principle

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.

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 Oblivion also provide an experience of weightlessness.


By 1907, Einstein had formulated the equivalence principle. He used it, together with special relativity, to predict that clocks would change rate in a uniform gravitational field. If clocks change rate, and special relativity holds locally to each clock, then distances must also be scaled. Hence the “shrinking” and “expansion” effect described as spacetime curvature. Einstein was at first defeated by the mathematical requirements. It took roughly another nine years, during which time Marcel Grossman taught him Riemannian geometry and tensors, before he published general relativity.

The Pound-Rebka Experiment

If clocks change speed in a gravitational field, in accordance with Einstein’s prediction, then this can be potentially be detected by means of redshift. In the Pound-Rebka experiment a Fe⁵⁷ radiation source was placed at the bottom of a 22.5 m (74ft) tube in the basement of the Jefferson Physics Laboratory at Harvard. Radiation of precisely the same frequency is likewise readily absorbed by Fe⁵⁷. The actual frequency of the radiation arriving in the penthouse at the top of the tube was determined using an absorber placed in motion so as to maximise absorption. Using the Doppler effect, the speed of motion of the absorber determines the shift in the frequency of the radiation. Pound and Rebka confirmed the prediction of relativity to an accuracy of better than 10%. The result was improved by Pound and Snider to better than 1%, and has since been confirmed using masers to better than 0.1%.

Gravitational Red Shift

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 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.

Mach’s Principle

It is known that Einstein thought very deeply about what he termed Mach’s principle prior to developing general relativity, but controversy has surrounded the question as to whether general relativity actually incorporates the principle, perhaps largely because it was never given clear expression. If the principle merely means that we can only talk of acceleration relative to other matter, then that is clearly the case in general relativity. However, the referenced matter is always local to the matter under consideration, and generally discussion of Mach’s principle seems to invoke a suggestion that rotation only makes sense in the context of the distribution of matter in the universe as a whole.

The origins of the discussion lie in Newton’s rotating bucket argument, that in the absence of absolute space it would not make sense to say that the bucket is rotating, and therefore no concave meniscus would form in its surface. Mach appears to suggest that the answer lies in the relative motion of the water in the bucket to distant stars. This idea is certainly not incorporated in the assumptions of general relativity, which is essentially a local theory. According to the restatement, N1*, of Newton’s first law, all we need consider is the motion of the particles of water relative to each other. The local structure of spacetime is determined from the interactions of particles locally, such that an inertial frame is one in which inertial particles can maintain a state of rest with respect to each other — effectively the situation when there is no meniscus signifying rotation of the bucket.

On the other hand a converse argument can be presented. If spacetime is divided into local, and overlapping, regions, each described in inertial coordinates, then no rotation may form in the global structure resulting by conjoining the regions. Thus we cannot say that the frame of the non-rotating bucket is determined from the “frame of the fixed stars”, but rather must say that the “frame of the fixed stars” is determined from local structures.

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