Einstein Relatively Easy

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The coordinate system from which an observer takes measurement of events in space and time (classical mechanic) or in spacetime (special and general relativity) is called a frame of reference.

Within the realm of Newtonian mechanics, an inertial frame or inertial reference frame, is one in which Newton's first law of motion is valid.

Newton's first law:

 

An object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. 

 

So then you might be wondering, when could Newton's first law ever not appear to be true?

Imagine you are on Earth and assume that you could remove both friction and air resistance.
Now hit a ball gently so it moves slowly along a perfectly smooth road with uniform velocity. If we were in a strict inertial frame, the ball would move in a straight line along the road, but it's not the case: due to the Earth's rotation, the ball's path is ever so slightly curved.

This force causing moving objects on the surface of the Earth to be deflected to the right (with respect to the direction of travel) in the North Hemisphere and to the left in the Southern Hemisphere, known as the Coriolis effect, along with the centrifugal force, is precisely what makes the rotating Earth a non inertial frame[1].

Coriolis effect in an inertial, at rest frame (top) and in a non-inertial, rotating (bottom) frame

More generally, apparent forces which are not caused by any physical interaction but are due to an observer using a non-inertial frame of reference - rotating or accelerating - are known as inertial or fictitious forces[2].

It leads to our second, more general definition of an inertial frame of reference:

Inertial frame of reference:

 

The inertial frame of reference is the one where the fictitious or inertial forces vanish. 

In other words, the laws of physics in the inertial frame are simpler because unnecessary forces are not present.

 

An inertial frame obey the following properties:

  1. If you are in an inertial frame and have no communication with the outside world, there is no experiment that will tell you whether your frame is at rest or moving with a uniform velocity: this is known as the Galilean relativity principle.
  2. Any frame that moves with constant velocity relative to another inertial frame is itself an inertial frame.
  3. Any frame that is accelerating or rotating relative to an inertial frame cannot itself be an inertial frame.
 
two frames of referential, with frame S' moving along the x axis of frame S with velocity v as measured in S.

Note 1:  In special relativity, inertial frames are known as Lorentz frames. In common with classical Newtonian inertial frames, they obey Newton's first law, but they differ in how they deal with gravity.
Unlike Newtonian inertial frames which treat gravity just like any other force, Lorentz frames can only be constructed in flat spacetime, known as Minkowski spacetime, one which is precisely not curved by the presence of mass/energy.

Note 2: As we will see in General Relativity course, the gravity field is equivalent to an non-inertial, uniformily accelerating frame, and the Earth's surface can be considered as accelerating up.
In this sense, the freely falling apple and not the Earth must constitute an inertial frame. In our article Geodesic equation and Christoffel symbols we show how gravity appears as an additional force due to nonuniform relative motion of two reference frames (free falling inertial reference of frame and accelerating up earth's frame). You can read this definition also of a Inertial Observer.

Note 3: concerning Earth taken as referential, one should distinguish between the Earth Centered Inertial (ECI) frames which have their origins at the center of mass of the Earth but dont rotate with it - so could be considered as inertial, and the Earth-centered, Earth fixed (ECEF) frames which rotate with the surface of the Earth

[1] You could find there a great video about Coriolis effect.

[2] Another commonly experienced example of inertial force would be the one that push you to the back of the seat in an accelerating car.

 

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"Five or six weeks elapsed between the conception of the idea for the special theory of relativity and the completion of the relevant publication" Einstein to Carl Seeling on March 11, 1952

"Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians."
David Hilbert

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