Inertial Frames in General Relativity

Today I don’t want to talk about code, but about one of my other passions: Physics…I want to start this new thread with something relative basic: the concept of inertial frame of reference. In physics courses the importance of such concept is often underestimated by teachers. The concept of inertial frames is not only important in Newtonian Mechanics, but even more important in Einstein relativity. Indeed such basic concept has been one of the leading cause that has lead Einstein and Minkowski to come up with the theory of relativity. But lets start from the begin…

In classical mechanics, an inertial frame of reference is defined as a frame of reference “whose net force acting upon it is zero”: in other words it is “not accelerating” . This simply means that the frame is either stationary OR moving at constant velocity in a straight line (indeed moving in a curved line will result an acceleration and thus not being inertial)… So why are so important in classical mechanics? Well because, you use them as reference to measure accelerations of other frames. Furthermore is worth to remark that, since the inertial frames do not accelerate, this automatically imply that all the inertial frames are not accelerating within each other. So if from one reference frame you find that another frame is accelerating, this means that one of the two is not inertial.. Also Newton’s second law F=ma is only true in inertial frame of reference.  The test for showing if a frame of reference is inertial or not is pretty simple. Pick up an object and release it: apart from the gravity, no other forces should act on the objects. In case the frame is not inertial (e.g. a car going around in circle) then the Newton second law should appear violated (e.g. the driver being “pushed/accelerated” sideways from no force). All this concepts are basic so nothing new until now…

Now lets consider four different frames:
– Frame 1: a box placed on the Earth surface
– Frame 2: a box placed out in free space far enough from any mater
– Frame 3: a box in free fall (dropped from an airplane)
– Frame 4: a box on-board of an  accelerating rocket in free space (where a=g).

According with Newton mechanics, the Frame 1 and 2 are inertial, while frame 3 and 4 are not inertial, because they actually accelerating. And exactly at this point that Einstein noted something simple, that became one of the postulate of the Relativity: The equivalence principle.
In particular he pointed out that from the prospective of an observer inside the boxes, he cannot distinguish between the Frame 2 and Frame 3 or between frame 1 and frame 4. Indeed any experiment that he tries to do (for example dropping and object) will lead to the same result (object floating from frame 2-3 comparison or object falling down with acceleration “g” between frame 1-4 comparison… ) .. So Einstein thought: what if actually Frame 2 and 3 are inertial frame of references and not Frame 1?  Well in order to claim that, he had to address some issues relative to the the Frame 3 being inertial. Indeed, since Earth is spherical and the box is falling towards the earth center, if we drop two apples they will tend to get closer to each other with no apparent reason or forces (because they will fall radially in the earth).  Also,  Orbiting frames of references (e.g. satellites) they also should be inertial because they are in “free fall”, but this is apparently wrong because they are accelerating respect to the Frame 1,2,3 and 4 (and inertial frame of reference should not accelerate respect to each other).. 

So after seven years, Einstein found the answer… The rule that inertial frames of references are not suppose to accelerate in respect to each other is only true in flat geometry. However, if the geometry is non-euclidean and curved then inertial  frame of reference can accelerate in respect to each other (due to concept of straight line in curved space)… Once Einstein figure out this the model of curved space time was born.

One remark about gravity… is it real? Well in Frame 4 the observer feel apparent force g because the rocket is actually accelerating. But what about Frame 1. Well for the equivalence principle,  the gravity effect is also “fake” and due by the fact that earth surface is accelerating outwards… Yes you read correctly.. From Relativity prospective, the earth surface act like is accelerating outwards. How is that possible? well we will need to dig more into the curved geometries and the geodesic for explaining this better…



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