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|18.2||Classical Principle of Relativity|
In this Physics tutorial, you will learn:
How many variables do we need to know in order to determine the position of an object at any instant when moving in space?
What are parametric equations in math? How many parametric equations can you use to describe the motion?
Do you think the velocity of a moving object is the same for all inertial frames of reference? (To answer this question consider two cars moving in opposite directions when viewed from the ground and from inside any of cars).
Do you think relativity can be applied in other fields of physics beside mechanics? Remember what happens when moving a magnet towards a coil to produce induced current in the coil. Can the same phenomenon occur when we move the coil toward a stationary magnet?
From your previous knowledge, can you respond whether is there anything invisible that fills the space of universe or not?
All these interesting but tricky questions will get answer in this tutorial, in which again, we will deal with classical theory of relativity, which act as a background for more advanced topics we will discuss in the next articles of this section.
Let's suppose there is an observer at rest, who is recording all events occurring in an inertial reference frame S. The origin O of space coordinates and the directions of the three axes X, Y and Z are all known. Using the standard methods, the observer associated to this inertial system measures the coordinates of any point of space. More precisely, he measures the space coordinates (x, y, z) of a material point which can be either at rest or at random motion in this inertial system.
From geometry, it is known that the 3-dimensional space is Euclidian. This means the distance of a point P(x, y, z) from origin O, is calculated through the Pythagorean Theorem.
As you see from the figure, L is the diagonal of the cuboid formed by projections of the point P on each axis and the point P itself. Hence, we can write:
In addition, the observer at rest in S measures the time elapsed through a standard clock. As discussed in the previous tutorial, the time is equal and absolute for all inertial frames of reference. Principally, we assume the observer as able to measure the time through a rigorously periodical process. We denote this time by t and take its origin (t = 0) at any instant, because time is homogenous.
From mathematics, it is known that a parametric equation define a group of quantities as functions of one or more independent variables called parameters. In our case, the time t is the parameter, so we can write general form of the parametric equations of motion as
where x, y and z are the coordinates of the point P at any instant t. Obviously, we know the object's motion when we know all the above three parametric functions at any given instant.
The diagonal L of the cuboid discussed above is usually denoted by r⃗ when expressed as a vector. It is known as position vector.
Now, let's consider the random motion of a particle P in a system S related to the Earth (i.e. we assume it as not moveable). The particle's trajectory is shown in the figure below. Then, we observe the same particle from another inertial reference system S'. Just we make sure (for convenience) to take the motion of the system S' in the X-direction of S. Obviously, the system S' moves at constant velocity V⃗ in respect to the fixed system S (as S' is inertial).
Let's take as t = 0 the instant in which the origins of the two system converge. It is clear that after a certain time t, the system S' is displaced by
in the positive direction of the X-axis. Therefore, at a given instant t, the particle P will be somewhere in the space and it will have a position vector r⃗ in the system S and r⃗' in the system S'.
From properties of vectors, is clear that:
For example, the system S' could have been connected to a train moving at constant velocity in respect to the ground. In the above figure, we also have denoted by v⃗ and v⃗' the instantaneous velocity of particle in the systems S and S' respectively.
Remark! Do not confuse the moving velocity V⃗ of the inertial reference frame S' in respect to the reference frame S (which is considered at rest) with the velocity of particle (v⃗ or v⃗') which represents the velocity of particle in the system S or S'. The first is denoted by an uppercase while the later with lowercase.
Giving that the three components of velocity vector V⃗ are (V, 0, 0), we obtain for the parametric equations of the point P in the system S' in respect to the system S:
When adding the time t as the fourth parameter and giving that t' = t in all inertial frames of reference, we obtain
These equations are nothing else but the equations of Galilean Transformations of Coordinates we obtained in the previous tutorial.
Let's wait until a short time interval Δt elapses after the particle have initially been at the position P, as shown in the figure discussed in the previous paragraph. Obviously, the particles coordinates have changed from x to x + Δx, from y to y + Δy and from z to z + Δz in the system S and from x' to x' + Δx', from y' to y' + Δy' and from z' to z' + Δz' in the system S'.
Subtracting side by side the corresponding quantities and considering the Galilean Transformations obtained earlier, we get
Dividing side by side all the above equations by Δt, we obtain:
The limit of the above equations for Δt → 0 give the corresponding velocities. Thus, we obtain
These formulae represent the three Galilean Transformations for Velocity. As we see, not only the position is relative; the velocity can be relative as well. Velocity depends on where do we make the observation (in S or S').
The above formulae represent the velocity in the system S' in respect to the system S. We can also express the inverse relationship between velocity components, i.e. the velocity in the system S in respect to values in the system S'. Thus, we have
An athlete running at 28.8 km/h throws a javelin in the forward direction. If the velocity of javelin relative to the athlete is 30 m/s, what is the horizontal component of javelin's velocity when it pins on the ground? Ignore the air resistance.
Since air resistance is not considered, we assume the horizontal component of all velocities as constant. When we consider a reference frame connected to the ground (at rest), we obtain for the horizontal velocity vx at which the javelin pins on the ground as
where V is the running velocity of the athlete and vx' is the javelin's velocity relative to the athlete.
Thus, giving that
we obtain for the horizontal component of velocity by which the javelin hits the ground:
As for the vertical component of javelin's velocity, it is not affected by what reference frame we choose, because the initial velocities (running velocity and javelin's throwing one) are both horizontal. Likewise, there is no change in the z-component of velocity (all are zero) regardless the reference frame we choose.
So far, we discussed about a number of inertial systems' features. Now it is clear that in inertial systems of reference, the Newton's Laws of Motion (including the Gravitational Law of Attraction) are applied. Moreover, we found that the transformation formulae regarding the coordinates and velocity when switching from one inertial reference system into another, contain the relative velocity V, which is the velocity by which an inertial system moves in respect to another inertial system (usually one of these systems is assumed at rest and the other as moving at constant velocity V).
However, it is easy to detect that the absolute velocity (v in the system S and v' in the system S') of a certain particle is irrelevant in the coordinates transformation formulae. Let's explain why. For this, we use again the figure of the first paragraph.
Let's suppose that at a certain time instant t, the particle has the velocity v⃗ (at S) and v⃗' (at S'). Now, we take a very small time interval Δt, similar to that taken when discussing the Galilean transformations of velocity. The particle will move from position P to a new position P'. Again, here the x-component of velocity in the two systems, S and S' has changed in the following way:
Giving that the relative velocity V between the two reference systems is constant, we obtain for the acceleration after dividing the above equations by the time interval Δt and taking the limit when ∆t → 0:
Therefore, we have
From the above equation, we conclude that the acceleration remains constant; no matter how the motion is made. Thus, we say that "acceleration (unlike position and velocity) is an absolute quantity (just like the time t) in all inertial systems of reference." In other words, acceleration in independent from the system it is measured for the same particle's motion.
On the other hand, the mass m of this particle does not change when switching from one inertial system of reference to another. Therefore, the force acting on the particle is constant as well, based on the Newton's Second Law of Motion, F = m · a. This means the force F, which describes the intensity of interaction and makes the particle deflect from the standard motion, is constant in inertial systems of reference - a conclusion we have also drawn in the previous tutorial. Therefore, the general equation of dynamics (2nd Newton's Law of Motion) and the Law of Inertia (1st Newton's Law of Motion) are the same for all inertial systems of reference.
From the Galilean Transformations prove that
where r12 is the position of a particle 2 in respect to another particle 1 in the system S and r12' is the same thing but in the system S'.
From the Galilean Transformations for coordinates, we have for the first particle:
and for the second particle:
From the equation of distance between two points:
All these facts are demonstrations for the full equivalence of all inertial systems in regard to mechanical (kinematic and dynamic) phenomena. From here, we attain the definition of the Classical Principle of Relativity:
All mechanical phenomena have the same features in all inertial frames (systems) of reference because the laws of mechanics have the same form for all of them.
Neither the absolute velocity nor the relative velocity of inertial systems to each other have no any role in mechanical-related phenomena. We can assume any of systems at rest and the other in uniform motion to it; we just have to use the proper system of equations derived from Galilean transformations for inertial systems.
We use to make all observations and calculations based on systems connected to Earth and therefore, we consider it as a system at rest. Suppose you are at a billiard hall playing with your friend. You see that many principles of dynamics like collisions, reflections, spinning, etc., are applied during the match. If you play the same game inside a ferry sailing at constant velocity on still water, all the above-mentioned principles of dynamics are still applied. Whatever value the velocity of ferry may have, no change in the laws governing the motion of balls is will occur; everything would happen as the event took place at the billiard hall on the ground we mentioned earlier.
Now, let's clarify an important point to avoid any confusion. Suppose one releases a ball from the window of a car moving at constant velocity. In the system S' connected to the car, the ball falls vertically down due to the pulling effect of gravity. However, if the same event is observed by another observer who is at rest in respect to the ground, he sees the trajectory of the ball is parabolic as shown in the figure below.
Is there any violation of the laws of mechanics in this case? Absolutely not! The Newton's Laws are applied in both systems. We already know that the only force acting on the ball is the downward force caused by gravity. Therefore, in absence of air resistance, the horizontal component of velocity is constant in both systems (0 in the system S' and vCar in the system S). Therefore, since the change in ball's trajectory comes due to the change in its horizontal component, there is no violation of the laws of mechanics in this case. Once again, we must stress that we are not able to say anything for the state of motion of a system if we are inside an inertial system of reference; we can assign any velocity to objects moving on it - even zero. Just we must pay attention the velocity do not exceed the ultimate speed (the speed of light in vacuum), for which we must discuss more extensively in the next paragraph.
An airplane has a velocity of 1000 km/h in still air. It is flying on a windy day, where the wind velocity is 20 km/h and is directed perpendicular to the moving direction of airplane. Calculate:
Earlier we saw that Newton's laws have the same form in all inertial systems. As a result, every observer sees the same mechanical phenomena that obey to these laws. The question that naturally arises here is: What happens to electromagnetic phenomena in different inertial systems of reference? In particular, what happens when electric and magnetic field-related processes are observed in two different inertial systems of reference? What happens to the EM waves and the propagation of light as an EM wave? All these questions led to the development of Einstein's Theory of Relativity (both special and general).
Since light is an EM wave, it contains both the electric and magnetic component. Giving that in SI units
we can find an equation for the speed of light in terms of the above constants based on the dimensional analysis, which is
The small deflection from the experimental value (299 792 km/s) comes because of the rounding made during the above calculations.
As discussed in tutorial 16.7 "Faraday's Law of Induction", when we move a magnet towards to or away from a coil, a current is induced in the coil. The same phenomenon is observed even if the coil itself is moved towards to or away from a fixed magnet (any change in magnetic flux produces an induced emf and current in the coil, no matter how this change in flux is obtained). We can call the system connected to the coil at rest as S and that connected to the coil in motion as S'. If the speed of coil relative to the magnet is equal in these systems, both of them produce the same amount of induced current.
From this example, it is clear that we can extend the relativistic reasoning used in mechanical-related phenomena in electromagnetic-related ones as well. The same reasoning can be used in many physical phenomenon that are not directly related to classical mechanics or electromagnetism. The first scientist who understood this symmetry of inertial systems, was Henri Poincare, who extended this approach for all physical-related phenomena, not only for the mechanical and electromagnetic ones. For this reason, Poincare is often considered as the true founder of Relativity.
However, some questions which seemed insurmountable at the first glance, did appear by the development of this theory. They consisted on the light propagation in vacuum, which albeit at very high speed, is a process that occurs at a finite speed (about 300 000 km/s). In the classical physics, it was inconceivable a wave process (such as the light propagation) can occur in vacuum. Scientists believed at that time that a material medium is necessary for the propagation of light waves, just like occurs in other types of waves (mechanical waves). They tried to explain this drawback by "inventing" the concept of "cosmic ether" - a medium extending at infinity, odorless, massless, transparent and non-observable in normal conditions. In this way, they believed that all objects flow through this cosmic ether without any mechanical resistance. It was supposed to be there only for allowing the conduction of EM waves from one place into another. Furthermore, the fictional cosmic ether was supposed to penetrate all objects. The two constants ε0 and μ0 were thought to be the electric and magnetic constants that characterize the cosmic ether while ε and μ the corresponding electric and magnetic constants characterizing the space inside material objects.
In the Newtonian system, the supposed cosmic ether was though as the absolute system of reference and all systems moving at constant velocity in the universe, were thought as inertial systems of reference inside this infinite absolute system. Thus, the speed of light in vacuum (about 300 000 km/s) was considered as the speed of light relative to the cosmic ether. The explanation give was that light speed has the same value in all directions because the cosmic ether is homogenous everywhere, so the concept of isotropy [uniformity in all directions] of light propagation was a common belief in the scientific circles.
However, in the above reasoning there is a nonsense. Let's suppose an inertial system moving at velocity V relative to the absolute system at rest represented by the cosmic ether. If we emit a light ray in the direction of V, the relative speed of light would be
When light is emitted in the opposite direction of V, the relative velocity of light would be
In this case, we would obtain two different values of light speed in the same medium. This would violate the principle of isotropy of cosmic ether, i.e. there is a paradox in this reasoning. Therefore, it remains that there is no cosmic ether in the universe, so the above theory is wrong. The correct explanation of relativistic phenomena was first provided by Einstein during the first decade of the last century; a theory we will discuss in the next tutorial.
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