The Problem of Slowing Clocks in Relativity Theory

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1 The Problem of Slowing Clocks in Relativity Theory The basic premise of Relativity Theory is that the speed of light ( c ) is a universal constant. Einstein evolved the Special Theory on the assumption that all observers would measure the same value for c no matter what their state of motion, with the limitation that this motion had to be inertial (constant velocity in a straight line). This led to the controversial conclusion that motion had to slow clocks and shrink measuring sticks (in the direction of motion) for that was the only way that observers in different inertial frames could find the same value for c. In the Special Theory, it is not clear that actual slowing of clocks is what is being asserted. Einstein writes that the clock in another frame moving past would appear to run slower to an observer at rest. However, it becomes clear in the General Theory that clocks in nature actually do slow down, according to Einstein. The General Theory is about gravity and he asserts that gravitational force will slow clocks. However, his equivalence principle states that the effects of gravity and acceleration are the same, so it is clear from the General Theory that acceleration, like gravity, will cause a clock to slow down if acceleration is equivalent to gravity. At this point, we should pause to consider that gravity and acceleration are not exactly equivalent. First, as Einstein pointed out, gravity has a different directional effect as smaller masses (e.g. satellites) will all be drawn toward the center of a large mass like a planet or star. If observers in a space station in Earth orbit drop two heavy weights from opposite sides of the station, the weights will start to come together as they fall because they will converge on the Earth s center. In contrast, an accelerating force will simply push a body in the same direction in a straight line, assuming no other intervening forces. Another important difference between acceleration and gravity is that there is no mutually attractive force in the case of acceleration, while the gravitational force is a function of the masses of both bodies since they are attracted to each other. This leads to a third difference which is that an orbiting body (like a space station or asteroid) will experience more acceleration from the gravitational force exerted by a massive body like a planet to the degree that the orbiting body itself is more massive because of the mutual attraction of the two masses as first explained by Newton. In contrast, there is no benefit to acceleration of a body in space assuming the absence of a gravitational field if the body has a larger mass. More force is required to achieve a given amount of acceleration if there is more mass. There is symmetry here in the sense that a more massive body will experience greater acceleration due to gravitational attraction and the same body will require more force to accelerate it compared, in each case, to a body with less mass. However, Einstein asserts that acceleration increases the mass of a body and the force required to move it at light speed would be infinite as a result. In contrast, the gravitational force between two bodies increases as the distance between their centers diminishes (Newton) but there is no increase in mass as a result of the acceleration of the smaller body due to gravitational attraction. Gravitational attraction cannot depend upon the respective masses of two bodies and at the same time increase those masses. Returning to the matter of slowing clocks, the formula for this usually referred to as the time dilation factor is derived in the Special Theory that deals with inertial frames. This same time dilation factor is

2 carried forward into the General Theory that deals with gravity and accelerated frames. I have pointed out before (see Internal Conflict in Relativity Theory ) that this seems to be a problem for relativity theory as a whole, and that the derivation of the formula goes against Einstein s own postulate in the Special Theory that light propagates independently of its source. That is because the light beam in a light clock situated in a rocket moving at close to light speed would have to be deflected forward in order to reach the mirror on the other side of the clock. But that should not happen if such beam proceeds independently in a straight line in space once it has been emitted, which is what Einstein s postulate asserts. This can be further illustrated by considering one of Einstein s more important thought experiments which led him to the principle of equivalence. If a light beam enters through the side window of a rocket proceeding through empty space at high velocity, it would be seen to move toward the rear of the ship before reaching the opposite side. If the rocket was travelling at constant velocity, the light beam would be seen to move in a straight line. If the rocket was being accelerated, the light beam would appear to pursue a curved course. Einstein then imagined an observer in a gravitational field suspended by a rope in an enclosure like an elevator. As a light beam entered the enclosure, it s path would curve toward the floor due to gravitational attraction. In this way, Einstein saw gravity and acceleration having an equivalent effect. But Einstein was so taken with this discovery that he did not carry the thought experiment further by asking a simple question: what if the light beam was emitted by a light source situated on the wall inside the rocket or the enclosure? Why should such a beam of light behave any differently from one emitted outside that came in through a window? If light propagates independently, once emitted it should travel in a straight line (absent gravity) and the motion of the emitting frame should not have any effect on it. Certainly this is true for the elevator as it is assumed to be at rest, suspended by its rope, and could not influence the direction of any light beam internal or external. Another way to look at this is to imagine a rocket travelling close to light speed that is prompted to emit a pulse of light from a signal lamp fixed to the side of the ship as it passes a given point in space that is opposite a light sensor some distance away. Would the emitted light beam strike the sensor or be deflected forward by the ship s motion and therefore miss the sensor? Most of us would assume that the light pulse, once emitted and independent of its source, would move in a straight line to the sensor and would not be deflected forward. But if this is so, the light clock exercise in the Special Theory where the time dilation formula is derived is invalid. The formula derived for time dilation in the Special Theory deals with the amount of time difference between two clocks after a given distance has been travelled. It does not address the matter of the run rate of an accelerated clock. This is evident from the formula itself for time dilation which is based on velocity: T = t x square root(1-v2/c2). And velocity here is the relative velocity observed between the clocks, not absolute velocity. Absolute velocity would have to assume forbidden concepts such as absolute rest or the Newtonian notion of fixed space. However, we can imagine measuring the speed of a given body travelling parallel to a light beam in empty space between two points as having an absolute velocity measured as some percentage of light speed. Of course the problem always arises of how to

3 synchronize clocks in these kinds of experiments, but at least we can imagine absolute velocity this way even if it cannot be precisely measured. In contrast, relative velocity ( V ) is meaningless for determining the change in run rate of a clock since it is only the perceived relative speed of approach or separation of two clocks in different inertial frames as measured by observers in those frames. (I have written that the idea of a negative value for velocity in Special Relativity is particularly meaningless as, conceptually, a body cannot have less than zero velocity.) And the Special Theory states that V cannot reach the speed of light even though a third observer in a different frame could see the clocks moving in opposite directions, each travelling at close to light speed (here meaning absolute velocity), and thus determine a relative velocity for the clocks of close to twice c. Further, two clocks not moving on parallel trajectories can have quite different absolute velocities from their observed approach / recession velocity. So relative velocity is different from absolute velocity and surely it is the latter that determines the slowing of a clock s run rate. So the time dilation factor from the Special Theory tells us nothing about the process of acceleration that resulted in a perceived relative velocity or, more importantly, the absolute velocity of whatever clock was accelerated. Here I am assuming based on the General Theory as pointed out above - that it is acceleration that causes a clock to slow, and that the run rate of a clock is not determined by how far it travels at a given velocity. (However, the amount of time difference will depend on the distance travelled since that requires an amount of elapsed time.) It is then logical to conclude that the time dilation factor derived in the Special Theory is not appropriate to use in the General Theory that focuses on gravity and acceleration, and where it is asserted that clocks actually do slow down under such conditions rather than just appearing to slow. The amount of slowing, then, ought to be related to the amount of force applied to the clock either by gravity or by some other agent acting to accelerate it. And clock slowing here is not a matter of relative perception but instead is an absolute physical effect. The most obvious experiment to show time dilation was the Hafele-Keating experiment of 1971 where atomic clocks were flown around the world on commercial airliners and brought back to compare with identical clocks kept at the U. S. Naval Observatory. Similar experiments have been performed since with greater accuracy. These experiments appear to prove Einstein s theory, just as other experiments have demonstrated the slowing of clocks due to gravitational attraction, as predicted in the General Theory. But the problem with Hafele-Keating and similar experiments is that they all depend upon the relative velocity of the respective clocks (although they do properly adjust for differing gravitational effects). The various accelerations and decelerations of the moving clocks and the related forces applied to these clocks are not considered. In fact, the idea that acceleration (or deceleration) is the cause of changes in the run rate of clocks is specifically rejected. Yet, as I have argued above, relative velocity is not something specific to any given body or clock that is moving. What a clock feels is the force acting on it to accelerate it or slow it down. Meanwhile, if gravity slows clocks and acceleration is supposed to have the same effect as gravity according to the General Theory then acceleration, like gravity, should slow the run rate of a clock. Hafele-Keating and similar experiments also fail to consider a third factor that is part of the General Theory, which is that acceleration increases mass. Since we know that the larger the (rest) mass the

4 greater the gravitational attraction, and that gravity slows the run rate of a clock, it should be assumed that the acceleration of the moving clocks in these experiments increased their mass which also increases the effect of gravity and causes further slowing, although this effect would be offset by the greater distance from the Earth s center achieved by flying at higher altitude. While this effect of mass gain due to acceleration was not considered in the clock experiments, I will be the first to admit that it would likely be so small as to not be measureable given the small rest mass of the clock. But what if we extrapolate the result of these experiments to more massive bodies in space moving at relativistic speeds? In such case, the increase in mass from acceleration and the resulting effect on gravitational attraction would be a material factor that would have to be considered. Einstein did not attempt to explain the actual physical process of clock slowing. In both gravitation and acceleration, force is applied to the clock. Intellectually, it is appealing to assume that the application of force causes processes to slow down. An alternative but related cause might be the increase in mass from the application of the accelerating force, but there does not seem to be a parallel with regard to gravity except that gravitational attraction is greater if mass is larger. The common factor between gravity and acceleration is the application of force to a clock, so that would seem to be the causal factor in slowing. Meanwhile, gravity can slow a clock without moving it to any velocity, relative or otherwise. So where is the parallel between relative velocity between moving and rest clocks as compared to clocks situated at different elevations? Certainly the higher clock will feel less gravitational force than the clock that is closer to the center of the Earth, while a moving clock will have felt the force that accelerated it to a given velocity (relative to a rest clock) as well as the force that causes it to reduce velocity and return to the positon of the rest clock. The moving clock will also feel additional force as it moves in a curved trajectory. This brings us to a final question. If the application of force to a clock causes it to slow, what is the effect of deceleration? Classical time dilation from the Special Theory appears to argue that the run rate of a clock (not the amount of elapsed time) depends upon achieving a certain absolute velocity and that using force to increase or reduce its velocity has no effect on the run rate. Time difference is based upon the elapsed time shown by two clocks that are separated by distance and brought back together again. To achieve that, at least one of them has to be accelerated to a relative velocity (in the twins experiment there will always be both approach and recession velocities and both accelerating and decelerating forces). What seems more logical to me, as I have argued before, is that the run rate of a clock is determined by its absolute velocity measured against a light beam. But this requires that while application of force to accelerate the clock will slow its run rate, the application of subsequent force to slow its absolute velocity will cause the clock s run rate to speed up. That seems counter-intuitive and yet we must assume from the General Theory that deceleration of a body reduces its mass, even though the same amount of force is applied to it to slow it down as was applied to accelerate it. Is the run rate of a clock - as well as its mass - tied to its absolute velocity (its speed relative to a parallel light beam) or is it a function of the forces applied to it, either gravitational or accelerating / braking?

5 In any event, relative velocity (that is only apparent according to the Special Theory) is not the proper cause of time dilation and is not equivalent to the gravitational slowing of clocks predicted in the General Theory. Meanwhile, acceleration in contrast to relative velocity is supposed to be equivalent to gravity according to the General Theory. Acceleration is always and everywhere a physical event (with cause and effect) while relative velocity is only apparent to observers in different reference frames where, Einstein says, neither of them can be sure which one is actually moving. C. Terrence Fisher January 2018 More Articles Millennium Relativity home page

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