Slide presentation Notes_03.ppt In the preceding slide presentation, we got to the point that detected no difference in the speed of light propagating in the same direction as Earth in its orbital motion, and in the opposite direction. There were to possible explanations: (a)Ether does exist, but for some reasons light does not obey the Galilean Transformation; and (b) There is no such thing as “cosmic ether”. At the end of XIX-th century the “ether theory” was so firmly embedded in physicists’ minds that for most of them option (b) seemed dubious, or even “heretical” (heresy is a dissent or deviation from a dominant theory, such as, e.g., a religious dogma). There were many attempts to find explanations of the results of Michelson-Morley experiments that would require the rejection of ether. One such explanation that gained much popularity and was supported by Michelson, was the “ether dragging” hypothesis (it was not new at that time, physicists had started speculating about possible “ether dragging effects” long before Michelson and Morley made their historic experiment.

Problems with classical relativity… About the same time when Michelson was performing his experiments, physicists became aware of one more puzzling fact. There was something wrong with Galilean Transformation… One fundamental principle that seemed quite obvious to everybody was the following: “Laws of physics should be the same in all inertial frames, even if they move relative to each other”. In other words, if we apply the Galilean Transformation to the equations expressing a given law, the equations should not change. This principle can be summarized in a single short sentence: The laws of physics should be invariant with respect to the Galilean Transformation. For instance: take Newton’s Second Law: We showed (see Notes_01.ppt, Slide 22) that the Galilean Transformation does not change the acceleration in a moving frame – so, the Second Law In this frame does not change, either! Equivalently, one can say: The Newton’s Second Law is invariant with Respect to the Galilean Transformation.

Problems with the Galilean Transformation… About the same time as Michelson failed to detect ether motion, other physicist noticed that the famous Maxwell Equations are NOT INVARIANT with respect to the Galilean Transformation! The transformation, with respect to which the Maxwell Equations were Found to invariant, was an “exotic” transformation formula discovered In 1905 by a Dutch physicist and mathematician, Hendrik Lorentz. Albert Einstein appears on the stage… Lorentz was a friend and mentor of Albert Einstein, a much younger physicist (in 1905, Einstein’s age, 26, was exactly one-half of Lorentz’s age). Einstein’s young age was probably a “favorable factor” in what happened next. Young people are courageous. And what Einstein did, indeed required a big dose of courage. Namely, in his historic 1905 paper Einstein formulated a “revolutionary” postulate: The speed of light is the same in all frames of reference, no matter what their own speed is.

Einstein’s revolutionary theory Einstein explained his revolutionary postulate by the idea that not only Maxwell Equations are invariant with respect to the Lorentz Transformation – it is all laws of physics that obey this transformation. But what about the Galilean Transformation – is it no longer valid?!! How comes? It does work well in all Newtonian mechanics, and in many other known physical phenomena… The Galilean Transformation, Einstein explained, is the limit of of the Lorentz Transformation when the relative speed u of the two frames of reference becomes very small: u → 0. But what does “very small” mean in practice? It means that u has to be much, much smaller than the speed of light c. The speed of light is approximately km/s, or miles/s. Think of fastest objects we can deal with on Earth or the outer space we travel to. How fast they are? Please answer. The fastest object the humanity has had a “close encounter” with since the beginning of our civilization was probably the “Tunguska Meteorite” that exploded over Siberia exactly 100 years ago. Its speed could be as high as 50 km/s (31 miles/s). 31 miles/s is still an extremely small number compared with c. So, in most conceivable “on-Earth” situations it is OK to use the Galilean Transformation.

Relativity of time To measure time, we use clocks. The hearth of any clock is a “pacemaker”. In the old “grandpa clocks” it was a swinging pendulum. When I was young, “quartz clocks”, in which an oscillating quartz crystal is the “pacemaker”, were huge and very expensive devices of the size of a refrigerator. Today, you can buy one for a few $$, and you carry it on your wrist. In atomic clocks – the most accurate existing clocks, one second per million years! – the “pacemaker” is a beam of Cesium atoms oscillating between two energy states. Well, but since the speed of light is constant, why shouldn’t we use LIGHT as a “pacemaker”?

“Light clock” One of the favorite Einstein’s “tools” he used in his considerations were “gedankenexperiments”, or “thought experiments” (in German, which was Einstein’s native tongue, “Gedanken” means “a thought”). Here is a definition from Wikipedia: A thought experiment (from the German Gedankenexperiment) is a proposal for an experiment that would test a hypothesis or theory but cannot actually be performed due to practical limitations; instead its purpose is to explore the potential consequences of the principle in question. Famous examples of thought experiments include Schroedinger's cat… The “light clock” is a gedankenexperi- ment tool. It consists of a flash bulb that sends a light pulse to a mirror, L 0 apart. When the back-reflected pulse reaches the detector D, it generates a signal that triggers another flash from the bulb. So, the bulb flashes at regular time intervals Δt = 2L 0 /c.

Light clock operation How the light clock works, it is shown in this animation – but look only at the first frame (we will look at the following frames in a moment). Click here: (1) relatvisticclocksrelatvisticclocks There are more such animations: (2) TimeDilationTimeDilation and (3) Now, there comes a very important moment. Suppose that two such light clocks are located in two reference frames, O and O’ (in the animation (3) the observer’s name in O is Jack, and in O’ is Jill). In both frames the bulb is located at the origin, and the back-reflecting mirror is on the Y axis (i. e., the vertical axis). When the two frames do not move relative to each other, each observer sees the same frequency of flashing in his/her own frame, and in the other frame.

The relativity of time Now comes a very important moment in our reasoning. Suppose that now the O and O’ frames move relative to each other Along the X (horizontal) axis with constant velocity u. Let’s look at this animation: relatvisticclocksrelatvisticclocks After clicking twice on “Next”, we will see how the observer in O sees the situation in his/her own frame (right side of the screen) and in the moving frame. The observer in O sees the clock in O operating in a “normal” way. However, the situation in O’ now looks different to him/her: the bulb in O’ flashes, but before the light signal reaches the mirror, the mirror travels some distance to the right. So, the observer in O sees that the light traveled along a slanted path – i.e., a longer path than in his/her “own” clock. And a similar thing happens to the back-reflected signal: before it reaches the detector, the detector moves some distance to the right, so the path traveled by the signal on the way back is again longer than in the O clock. If you click on “Next” one more time, the program will display the paths traveled by the light signals in both frames – let’s stress: as seen by the observer in the O frame.

Continued from the preceding slide: Now comes the moment when we have to use the Einstein’s postulate that the speed of light is the same and equal c for observers in all frames! The observer in the O frame “saw” that the light signal in O’ traveled along the slanted paths shown by red lines in the animation. Suppose that the time interval between the two flashes in O’ he/she registered was Δt’. So, the length of each slanted path section was cΔt’/2. And, for the O observer, between the two flashes the (bulb+detector) unit in O’ moved by a distance uΔt’ along the X axis. From the line plot in the animation, one can readily see that: From which we obtain:

Continued from the preceding slide (2): After more manipulations: So, this is the time interval between two flashes in the O’ frame that the observer in O’ registers. But his/her “own” clock flashes at time Intervals: And both clocks, let’s stress it are identical!! Conclusion?

Continued from the preceding slide (3): The conclusion is that: (“always” = if the frames are in motion relative to each other) it means that observer O sees that the clock in O’ runs slower than in O – in other words, time passes slower in O’. Now, there is an important question: does then the observer in O’ see that the clock in O runs FASTER than his/her “own” clock? Answer: No, the observer in O’ “sees” exactly the same as the O’ observer – it is his/her clock that runs faster than the clock in O. Does it make sense? Definitely, it is difficult to answer enthusiastically “Yes, of course!”