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 galileo_theory_relativity [2020/05/17 00:06]admin galileo_theory_relativity [2020/05/17 00:10] (current)admin Both sides previous revision Previous revision 2020/05/17 00:10 admin 2020/05/17 00:06 admin 2020/05/17 00:05 admin created 2020/05/17 00:10 admin 2020/05/17 00:06 admin 2020/05/17 00:05 admin created Line 1: Line 1: + ====== Galileo'​s Theory Of Relativity ====== + [[books:​dialogues_concerning_two_new_sciences_by_galileo_galilei|]] + + <​html>​ + Imagine a person inside a ship which is sailing on a perfectly smooth ​ + lake at constant speed. This passeneger is in the ship's windowless ​ + hull and, despite it being a fine day, is engaged in doing mechanical ​ + experiments (such as studying the behavior of pendula and the + trajectories of falling bodies). A simple question one can ask of this + researcher is whether she can determine that the ship is moving (with + respect to the lake shore) <​em>​without going on deck or looking out a + porthole​.

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+ Since the ship is moving at constant speed and direction she will not <​em>​feel​ + the motion of the ship. This is the same situation as when flying on a + plane: one cannot tell, without looking out one of the windows, that + the plane is moving once it reaches cruising altitutde (at which point + the plane is flying at constant speed and direction). Still one might + wonder whether the experiments being done in the ship's hull will give + some indication of the its motion. Based on his experiments Galileo ​ + concluded that this is in fact impossible: all mechanical experiments ​ + done inside a ship moving at constant speed in a constant direction ​ + would give precisely the same results as similar experiments done on + shore.

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+ The conclusion is that one observer in a house by the shore and another ​ + in the ship will not be able to determine that the ship is moving by + comparing the results of experiments done inside the house and ship. In + order to determine motion these observers must look at each other. It + is important important to note that this is true <​em>​only​ if the + ship is sailing at constant speed and direction, should it speed up, + slow down or turn the researcher inside <​em>​can​ tell that the ship + is moving. For example, if the ship turns you can see all things ​ + hanging from the roof (such as a lamp) tilting with respect to the + floor

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+ Generalizing these observations Galileo postulated his <​b>​relativity ​ + hypothesis:

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+ <​center>​ + ​ + <​tbody><​tr>​ + <​td><​em>​any two observers moving at constant speed and + direction with respect to one another will obtain the same results for + all mechanical experiments ​ + + ​ + ​ +

+ (it is understood that the apparatuses they use for these experiments ​ + move with them).

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+ In pursuing these ideas Galileo used the scientific method (Sec. ​1.2.1​):​ he derived consequences ​ + of this hypothesis and determined whether they agree with the + predictions.

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+ This idea has a very important consequence:​ <​em><​font color="#​00ff00">​velocity ​ + is not absolute​. This means that velocity can only be + measured in reference to some object(s), and that the result of this + measurment changes if we decide to measure the velocity with respect to + a diferent refernce point(s). Imagine an observer traveling inside a + windowless spaceship moving away from the sun at constant velocity. ​ + Galileo asserted that there are no mechanical experiments that can be + made inside the rocket that will tell the occupants that the rocket is + moving . The question are we moving''​ has no meaning unless we + specify a reference frame (are we moving with respect to that star''​ <​em>​is​ + ​meaningful). This fact, formulated in the 1600's remains very true + today and is one of the cornerstones of Einstein'​s theories of + relativity.

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<​img src="​https://​www.ganino.com/​_media/​timg143.gif"​ alt="​\begin{figure} \framebox [6 in][r]{\parbox[r]{5.5 in}{\scriptsize \bigskip{\em ...  ...f experience. Think of it.\bigskip} \parbox{.2in}{\hspace{.1in}}} {}\end{figure}">​ + <​br>​ + <​p>​ +

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+ A concept associated with these ideas is the one of a frame of + reference''​. We intuitively know that the position of a small body + relative to a reference point is determined by three numbers. Indeed ​ + consider three long rods at 90<​sup><​i>​o​ from one another, the + position of an object is uniquely determined by the distance along each + of the corresponding three directions one must travel in order to get + from the point where the rods join to the object (Fig.
​4.1​)

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+ <​div align="​CENTER"><​a name="​1833">&​nbsp;​ <​p>​ + <​table>​ + <​caption><​strong>​Figure 4.1:​ A frame of reference.
&​nbsp;​ + <​tbody><​tr>​ + <​td><​img src="​https://​www.ganino.com/​_media/​img144.gif"​ alt="​\begin{figure} \centerline{ \vbox to 3 truein{\epsfysize=6 truein\epsfbox[0 0 612 792]{4.galileo/​f2.ps}} }\end{figure}"​ sgi_src="/​usr/​people/​wudka/​public_html/​Physics7/​Notes_www/​img144.gif"​ height="​313"​ width="​314">​ + + ​ +

+ <​br>​ + <​p>​ +

+ Thus anyone can determine positions and, if he/she carries clocks, ​ + motion of particles accurately by using these rods and good clocks. ​ + This set of rods and clock is called a <​i>​reference frame​. In + short: ​a reference frame determines the where and + when of anything with respect to a reference point.

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+ A prediction of Galileo'​s principle of relativity is that free objects ​ + will move in straight lines at constant speed. A free object does not + suffer form interactions from other bodies or agencies, so if it is at + one time at rest in some reference frame, it will remain at rest + forever in this frame. Now, imagine observing the body form another ​ + reference frame moving at constant speed and direction with respect to + the first. In this second frame the free body is seen to move at + constant speed and (opposite) direction. Still nothing has been done to + the body itself, we are merely looking at it from another reference ​ + frame. So, in one frame the body is stationary, in another frame it + moves at constant speed and direction. On the other hand if the body is + influenced by something or other it will change its motion by speeding ​ + up, slowing down or turning. In this case either speed or direction are + not constant as observed in <​em>​any​ reference frame. From these + arguments Galileo concluded that free bodies are uniquely characterized ​ + by moving at constant speed (which might be zero) and direction.

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+ An interesting sideline about Galilean relativity is the following. Up + to that time the perennial question was, what kept a body moving? ​ + Galileo realized that this was the <​i>​wrong question,​ since uniform ​ + motion in a straight line is not an absolute concept. The right + question is, what keeps a body from moving uniformly in a straight ​ + line? The answer to that is forces''​ (which are defined by these + statements). This illustrates a big problem in physics, we have at our + disposal all the answers (Nature is before us), but only when the right + questions are asked the regularity of the answers before us becomes ​ + apparent. Einstein was able to ask a different set of questions and + this lead to perhaps the most beautiful insights into the workings of + Nature that have been obtained.

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+ Galilean relativity predicts that free motion is in a straight line at + constant speed. This important conclusion cannot be accepted without ​ + experimental evidence. Though everyday experience seems to contradict ​ + this conclusion (for example, if we kick a ball, it will eventually ​ + stop), Galileo realized that this is due to the fact that in such + motions the objects are <​em>​not​ left alone: they are affected by + friction. He then performed a series of experiments in which he + determined that frictionless motion would indeed be in a straight line + at constant speed. Consider a ball rolling in a smooth bowl (Fig. ​4.2​).

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+ <​div align="​CENTER"><​a name="​1835">&​nbsp;​ <​p>​ + <​table>​ + <​caption><​strong>​Figure 4.2:​ Illustration of Galileo'​s ​ + experiments with friction
&​nbsp;​ + <​tbody><​tr>​ + <​td><​img src="​https://​www.ganino.com/​_media/​img145.gif"​ alt="​\begin{figure} \centerline{ \vbox to 2 truein{\epsfysize=8 truein\epsfbox[0 -70 612 722]{4.galileo/​f1.ps}} }\end{figure}">​ + + ​ +

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+ The ball rolls from it's release point to the opposite end and back to + a certain place slightly below the initial point. As the surfaces of + the bowl and ball are made smoother and smoother the ball returns to a + point closer and closer to the initial one. In the limit of zero + friction, he concluded, the ball would endlessly go back and forth in + this bowl.

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+ Following this reasoning and abstracting away''​ frictional effects he + concluded that

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+ <​center>​ + ​ + <​tbody><​tr>​ + <​td><​em><​font color="#​00ff00">​Free horizontal motion is + constant in speed and direction.<​font color="​green">​ ​ + + ​ + ​ +

+ This directly contradicts the Aristotelian philosophy which claimed ​ + that

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• + all objects on Earth, being imperfect, will naturally slow down, + <​li>​ + that in a vacuum infinite speeds would ensue, ​ + <​li>​ + and that perfect celestial bodies must move in circles. ​ + ​ +

+ In fact objects on Earth slow down due to friction, an object at rest + would stay at rest even if in vacuum, and celestial bodies, as anything ​ + else, move in a straight line at constant speed or remain at rest + unless acted by forces.

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