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Relativity. Galilean Transformations. Einstein's Postulates and Newton's Laws Revision Notes

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18.1Relativity. Galilean Transformations. Einstein's Postulates and Newton's Laws


In these revision notes for Relativity. Galilean Transformations. Einstein's Postulates and Newton's Laws, we cover the following key points:

  • What is an inertial frame of reference?
  • What does the Galilean Principle of Relativity say about motion?
  • What are the Galilean transformations? What are the equations expressing them?
  • What do the Einstein Postulates say on relativity?
  • What is an event is relativity?
  • What do the spacetime coordinates in the Newtonian System represent?
  • How do the Newton's Laws apply in inertial frames of reference?
  • What are non-inertial frames of reference?
  • What are some inertial systems of reference we can choose when studying the motion of particles? Which is the best one?

Relativity. Galilean Transformations. Einstein's Postulates and Newton's Laws Revision Notes

In physics, a frame of reference is an arbitrary set of axes used to determine the position of an object or the physical laws that govern its motion.

Inertial frames of reference are three-dimensional coordinate systems, which travel at constant velocity. In such frames, an object is observed to have no acceleration when no forces are acting on it. If a reference frame moves with constant velocity relative to another inertial reference frame, it represents an inertial reference frame as well. There is no absolute inertial reference frame; this means there is no state of velocity that is special in the universe. All inertial reference frames are equivalent. One can only detect the relative motion of one inertial reference frame to another.

The term "inertial" derives from the concept of "inertia,/b>" discussed in the Newton's First Law of Motion, which implies that an object moves at constant velocity unless an unbalanced force acts on it.

The non-priority of an inertial frame of reference to another inertial one constitutes the Galilean Principle of Relativity, formulated in 1635, long before Einstein generalized the concept of relativity by including non-inertial reference frames as well.

Mathematically, the Galilean Principle of Relativity expresses the invariance of mechanical equations with respect to transformations occurring in the coordinates of moving points (and time) when there is a transition from one inertial system into another. This means we have four variables included in such situations: the three spatial coordinates x, y, z and the time t.

To determine the object's coordinates in the inertial frame, S', when we know its coordinates in the original inertial frame S, we employ the Galilean space and time transformations. If S' has a velocity relative to S so that v' = 0, then we have:

x' = x + v ∙ t
y' = y
z' = z
t' = t

In Galilean transformations, time is the same in all inertial frames.

Einstein based his Special Theory of Relativity upon two postulated:

  1. The laws of physics are the same and can be expressed in their simplest form in all inertial frames of reference (we discussed this point earlier in this tutorial). This is known as the relativity postulate. The laws of physics mentioned in this postulate include only those that satisfy this postulate.
  2. The speed of light in vacuum has the same value c in all directions and in all inertial reference frames. This is known as the speed of light postulate.

The second postulate implies that in universe, there is an ultimate speed c, which is the same in all directions and in all inertial frames of reference. Light is the only known thing that travels at this ultimate speed. No entity that carries energy or information can exceed this limit. Moreover, no particle that has mass can actually reach the ultimate speed c, no matter how much or for how long that particle is accelerated.

Precise measurements using modern devices have given the value c = 299,792,458 km/s for the speed of light in vacuum.

An event is fully described when we know four parameters of it: the three space coordinates (x, y, z) and the time (t). These are known as the spacetime coordinates because in relativity, space and time are entangled with each other.

Newton believed in the existence of an absolute space and an absolute time entirely independent from physical processes and from the existence of material objects. If a particle moves uniformly in deep space, this is known as "standard motion". The particle in question does not interact with anything else, given that it is isolated in the absolute space. For this reason, it is called a "free particle" and the system in which this particle is moving is called the "absolute inertial system of reference." Every other system that moves in standard mode to the absolute system of reference is also inertial.

If an inertial system S moves at constant velocity v in respect to the absolute system S0, then the conditions for the Newton's First Law of Motion are met. This is because this law includes either situations where the objects at rest when they move at constant velocity (if no force is acting on the object, it either remains at rest or moves at constant velocity).

If a particle in an inertial system of reference is accelerated, i.e. if it does not perform a standard motion, we say the particle interacts. A force F - which measures the intensity of interaction - acts on the particle. On the other hand, the acceleration a the particle experiences, gives the intensity of particle's deflection from the standard motion. The relationship between these two quantities obeys the Newton's Second Law of Motion

a = F/m

Thus, the acceleration a particle experiences due to the action of a force is proportional to the force itself. This statement represents the Newton's Second Law of Motion expressed in words. Here, 1/m is the coefficient of proportionality, which turns the given proportion into equation and the mass itself expresses the inertia, i.e. the tendency of an object to resist to any change in its state of motion.

The Newton's Third Law of Motion (the action-reaction principle) completes the framework of Newtonian Dynamics. It says: "For every action, there is an equal-size but opposite reaction".

In this way, it is clear that when the net force acting on an object is zero, the object moves in standard mode (i.e. at constant velocity). Since it is practically impossible to find any free particle in Earth conditions, we can obtain inertial systems of reference only if the net force on an object is zero, as this situation is more realistic.

An inertial connected to a spacecraft moving into the interstellar space with the engines tuned off represents the best model of an inertial system.

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