Let us look at the first two laws. The first law says that in the absence of forces all bodies move along straight lines with constant speed. Straight lines are not a problem. A body can move in straight lines in spacetime as well. For any observer such a line represents a straight line motion in space as well as a function of the time defined by that observer. And all observers will see a straight line as a straight line even if they are moving. The uniform speed can be made trivial. There is a time for each observer which is the length of the path of the particle-- the proper time. The speed is the distance moved in a unit of time, bu the distance in spacetime along the path of the particle IS exactly the proper time. Thus all bodies move a unit distance in a unit of time. Ie, all bodies have a speed of 1, which is certainly uniform. Ie, straight lines in spacetime are automatically "constant speed" in terms of the proper time and the spacetime distance.
Since the speed of a body along any path (not necessarily a straight line) in spacetime is also unity, the only thing a force can do is to change the direction of motion of a particle. Essentially all motion is "centrifugal" motion in the sense of Huygens. All motion can do is to change the direction in spacetime. This means that there can never be a component of the force that points along the path of the particle. All forces are perpendicular (in the sense of Minkowski) to the spacetime path of the particle. This requirement limits the types of forces that one can have in spacetime, but does not limit the motions.
However the third law is problematic. The third law says that the force that body A exerts on body B is equal and opposite to the force that B exerts on A. But the comments about the second law above makes this difficult. The force must be perpendicular to the path of the particle, but the direction of motion of A and B can be different. This would mean that the direction of the force must be different. But that means it cannot be equal and opposite.
The second problem is that of asking when are the forces equal an opposite. The force can change along the path of the particle (change in time) and in Newtonian theory, the third law applies to forces at the same time. It is not that the force of A on B for all times must equal the force of B on A right now. But in special relativity, simultaneity is broken. Different observers will have different ideas of when "now" is. If one chose a specific observer to define the notion of "Now" when the forces are equal and opposite, then relativity would be broken. There would be a preferred observer in the universe-- the one for whom Newton's third law is valid.
If on the other hand, the two particles interact only when they are in contact, then during the contact, the two have the same path (so the direction of the force is the same), and their notion of simultaneous is the same. But this is clearly not a sufficient answer, since we know lots of interactions between particles (eg gravity) where the particles are not at the same place. It was Maxwell, that unwittingly gave the answer. If there are fields, then the particles can interact with the fields at the location of the particle. The field can then transmit that interaction to the other particle and interact with it at that other location. Of course now the problem is what the meaning is of a force acting on the field. Fields are not things which you can imagine a force acting on it.
What happened instead is that one of the consequence of the second and third laws was applied, the notion of the conservation of energy and momentum. The momentum is, in Newtonian physics, the mass times the velocity. In the 4-dimensional Minkowski spacetime, the momentum is also defined as the mass (in this case the so called rest mass) times the 4-velocity. Since the length-squared of the four velocity is 1, the length-squared of the 4-momentum is the just the rest-mass squared, a constant for that particle. In collisions, the 4-momentum is conserved, and one can take that as the new definition of Newton's third law.
One can also define energy and momentum for the field (eg the electromagnetic field) The expression is not particularly illuminating. The energy density (energy per unit volume) for the electromagnetic field is equal to a constant called the dielectric constant times the Electric field squared, plus a magnetic constant times the magnetic field squared. Similarly one can define a momentum density as a certain product (called the cross product) of the electric field times the magnetic field. Now one replaces Newton's third law by conservation of the 4-momentum of the particle, plus the 4-momentum of the field (the integration of the field's energy and momentum density over all space) is always constant.
The field's changes in the energy and momentum density, when the particle interacts with it, propagate at most at the velocity of light. Thus, the interactions with particles gets to them only at the velocity of light.
Once on got to quantum mechanics, it turned out that it is very difficult to make the individual particles, even in their interactions with the field, produce a reasonable mathematical framework. Instead one made everything into a field, like electromagnetism. Quantum mechanics made if possible for those fields to behave like particles in certain situations, explaining why things sometimes behaved like particles. But the theory was now a more-or-less self consistent structure, and one could retain a self consistent mathematization of the natural world.
Except gravity. By 1908 (the same year when Minkowski introduced the notion of spacetime and Einstein denigrated the idea) he began to worry about how Newtonian physics of gravity could be brought into the special relativistic framework. There was no field in Newtonian gravity. There were forces at a distance. One could make something like a field with what is called the potential for the Newtonian gravity (This is basically now much motional/kinetic energy the particle would pick up if you let is travel from some fixed point to the point of interest when guided by a frictionless track, like Galileo's inclined planes. In Newtonian physics, this changed only if the objects creating the gravity moved, and the changes were instantaneous everywhere (in order to make sure that Newton's third law applied). Clearly special relativity did not allow instantaneous effects (In whose frame should they be instantaneous, since simultaneity was different for different observers). His key notion was what has been called the Einstein elevator experiment. Given a closed box, and either place it on the surface of the earth or put it in space and accelerate it upward at 9.8meters/second squared. Is there any way that the inhabitant could tell the difference? In both cases objects would fall, but for different reasons. In the grounded room, one would say they fell because the gravity of the earth pulled it down. In the elevator, one would say they seemed to fall because the floor of the elevator was accelerating up and when your released the object, it would continue in a straight line with uniform velocity, but the floor being accelerated up would rise to meet the object. Explanation different, result the same. Similarly, the pressure under your feet in the two cases, in the one, gravity pulls you down, and the floor has to exert a pressure on you ( and thus you exert on on the floor) in order to prevent you from falling through the floor. In the elevator, since you have to be accelerating up with the elevator, it has to push on you to give you that accelerations. Thus, what we usually call gravity is just because have put ourselves into a situation where we are being accelerated-- deflected from following a straight line in spacetime.
It was now that Einstein grabbed Minkowski's idea of spacetime, and of distances in spacetime. What is a straight line? It is the shortest (or in spacetime for timelike curves, the longest) distance between two points. iP>If we could alter the notion of distances in the Minkowski spacetime, perhaps we could alter the notion of what those straight lines are, such that a person sitting on the earth does not follow a straight line. An example of this is to look at a beach ball on which one draws two lines a constant distance apart. Those are supposed to represent the two sides of the earth (say Vancouver and Isle Crozet in the south western Indian Ocean which are approximately antipodal). Now draw a straight line on the surface which leaves one of circles. It will eventually come back to that circle. representing say Vancouver. this represents the path of a ball thrown up away from the surface and it eventually comes back down to the surface.
A ball thrown up comes back down.
Ie, by altering the notion of distances on the surface of the earth one gets things to behave as if they were in a gravitational field. It took him about 7 years of floundering before, in Nov 1915, he got a theory he was happy with based on these principles.
The dominant effect for objects which travel at speeds less than the velocity of light, is that time is not something which is the same from place to place. In special relativity time changed with motion. But in general relativity, time changes from place to place, even if there is no motion between the places. Time runs slightly slower near the surface of the earth than it does far away from the surface of the earth, by an amount of about 7 10^(-10) (about 21 milli-seconds/year) slower than at very far from the earth.
This has been measured. Robert Vessot launched a rocket to a height 10000km above the earth, and had it telemetry back the readings from a Hydrogen Maser clock on board the spacecraft. He found that the clock in the rocket ticked faster (by about on part is .4 billionths) and agreed with Einstein gravity prediction to about .02%. Now a-days you can buy a clock for thousand dollars dollars and measure the different in the time rate at the base and the top of mountain. Colorado measurement Pike's Peak And for much less than a million dollars, the change in time has even been measured over a height difference between two clocks of less than a milli-meter.
It is often said that gravity causes clocks to go slower near a gravitating body. This is wrong. The clocks going slower IS gravity. Objects travelling in straight lines in a spacetime which has time going slower near the sun follow just the paths that Kepler found that the planets followed, and Newton explained with his inverse squared force. These planets have no forces on them. they are not accelerating. They are following the law of inertia of travelling along straight lines (maximal spacetime distance between two points).
This difference in distances also causes a deflection of straight lines. For anything but light, this spatial deflection is far smaller than the temporal deflection. For light they are the same, and in the same direction. Thus the light is deflected by twice as much as the above Newtonian amount (about 1.5 sec of arc for light just grazing the edge of the sun).
In 1919 an expedition was sent to a solar eclipse which went from South America just below the equator to Africa. It was led by F Dyson, the Astronomer Royal of Britain, and A Eddington a bright young theorist who had stayed out of military service because he was Quaker ( and thus a pacifist).
One half under Dyson want to Sobral, just inland from the coast of Brazil and south of Belem. The other half under Dyson went to a small island, Principe, off the coast of Guinea and part of the nation of Sao Tome and Principe.
When the eclipse occurred, storm hit Principe in the morning. During the eclipse there were still thin layers of high cloud which cleared somewhat toward the end of the eclipse. The weather at Sobral was much better. Unfortunately, they decided not to point the telescopes at the eclipse because of the difficulty of constructing an equatorial mount for a 10 foot telescope in Sobral. Instead they put a mirror at the front of the telescope which reflected the area of the sky where the eclipse was to occur into the telescope, and the mirror moved so as to compensate for the motion of the sky caused by the rotation of the earth. This worked fine before and after the eclipse, but the mirror seemed to judder during the eclipse. This made the pictures of the sky much worse than they could have been. They had a spare smaller lensed telescope as a backup, and it worked well. Thus only one of the three telescopes they had in the expedition worked well. The result from that telescope was that there was a measured deflection of about 1.9 seconds of arc at the limb of the sun, definitely excluding the Newtonian value, and even a touch big for the Einstein value. See this paper by Kennefick who analyses the accusation that Eddington was biased for Einstein, and made sure that the reported result supported him. Kennefick finds that the information does not support this accusation.
The deflection has been measured since, especially in radio frequencies when a radio noise quasar is eclipsed by the sun. You can "see" that even if there is no eclipse) and the deflection agrees with Einstein theory to about .02%. The Hipparchus satellite, whose purpose was to measure the position of many thousands of stars to better than .1 micro-arcsecond all over the sky. That measured the deflection of the starlight by the sun even for stars which are 90 degrees in the sky away from the sun.
Mercury is the planet closest to the sun, and also has a large eccentricity (about .2-- recall that the eccentricity is the ratio of he distance between the two foci of the ellipse to the length of the major axis of the ellipse) which means that one can measure accurately when the planet is closest to the sun. This is called the perihelion (near sun from Ancient Greek). In Kepler's theory, the direction, with respect to the stars, the line from the sun to the perihelion always points in the same direction. In Newtonian theory, while it always points in the same direction if one only takes account of the sun, if you take into account the gravitational effect of especially Jupiter, but also of Venus and Earth, Mercury's orbit is slightly distorted, and more importantly the direction of the perihelion very slowly rotates around the the sun. Throughout the 18th and 19th centuries immense time and effort was expended on these calculations (they for example led to discover of Uranus and Neptune due to the perturbations they caused to the orbits of Saturn and Uranus). They found that the other planets procured an advance of the perihelion of Mercury of about 1.5 degrees per century. But they differed from the measured value by about 40 seconds of arc a century.
This discrepancy was a major puzzle. After the discovery of Neptune, one hypothesis was that there was another planet very close to the sun -- so close that it was always lost in the glare of the sun-- and that was what made Mercury's orbit rotate by that 40 sec per century. This planet was even given a name, Vulcan (Vulcan being the Roman blacksmith god, who was constantly close to fire). One observer even claimed to have seen its crossing the sun. but noone else ever saw anything of the planet.
When Einstein calculated the orbit of Mercury in his theory, he found that, although it followed a path very close that calculated in Newton's theory, it differed in that the direction of the perihelion of Mercury was not constant but advanced by 43 seconds of arc per century. He did this calculation immediately after he had finished the theory and published it. In fact he had calculated the perihelion advance of Mercury in every theory he had tried in those 7 years, and the gross disagreement was one of the reasons (but not the crucial one) that led him to keep looking.
See this site which discusses some of the tests including an animation (grossly exaggerated by increasing the eccentricity from about .2 to .9, and the perihelion advance from 43 seconds per century to about 5 degrees per orbit.
In 1974 Hulse and Taylor found a binary pulsar system (two neutron stars orbit each other) with an 8 hour period. A neutron star is star with just a bit more mass of the sun but having collapse to a radius of about 10km-- the size of Vancouver. Compare the orbital period to that of mercury of about 80 days. The periaston advance of this system is about 4 degrees per year, rather than 43 seconds of arc per century, in agreement with General Relativity.
The double binary pulsar discovered in 2004 PS J0737-3039A/B has two pulsars (Neutron stars which emit a pulse of radio waves each time they rotate). The pulses form an incredibly accurate clock. The orbit of the two around each other is almost exactly edge on to us, so that, seen from earth the pulsars are almost exactly in line twice in the orbit. This means that the pulses from pulsar A (which we can see always, unlike pulsar B whose pulses seem to have been blocked throughout about 1/2 of its orbit) travel very close to pulsar B. This means that the Shapiro time delay is very large and easily measured.
See this second plot which shows the delay of the pulses of the pulsar as it goes in line, and almost behind the other one. The red line is the prediction from the theory. (The first graph is trying to predict the best fit if you neglect the Shapiro time delay, and is the difference between the measured values and the theory. It does not do well in the fitting) See Article by Cliff Will for a detailed list of the tests of General Relativity by 2014.
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copyright W Unruh (2018)