Newton’s theories stand up amidst obvious contradictions – An introduction to Einstein

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Newton (“Principia”, 1687) has a quite intuitive idea of the universe, related to the principles of the Euclidean geometry. In fact, bodies are imbedded into an empty space, endless and Euclidean, absolute time, that can be measured by Cartesian co-ordinates. Gravitational actions are instantaneous and originate from forces. 

By inertia a body, not subject to forces, moves following a rectilinear uniform motion. Universal gravitation is valid among reference marks that move, each other, following an uniform relative motion, only when an absolute time is taken into consideration. 

In the case of accelerated reference marks, “fictitious forces” become necessary to turn out matters. 

Cassini verified delays and advances during the transit of the satellite “Io”, crossing Jupiter’s shadow. A possible cause could have been the limited velocity of light. He doubted and, so, missed this discovery. His assistant, Römer, completed the job. 

Römer (1678) definitively proved that the velocity of light is finite. A valid estimate of its numerical value was achieved by his scheme of measure. 

Maxwell (1873) defined the equations that combined the magnetic field with the electric field. The velocity of propagation of the wave must always be the same and is related to constants that can be physically measured.

  = magnetic permeability in vacuum ----  = electric permittivity in vacuum 

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Maxwell’s theories stand up amidst obvious contradictions - An introduction to Einstein

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It is assumable that the possible point of reference for these equations must be jointed to the fixed stars. It is, however, immediately clear, that this is just an expedient. As a matter of fact, Maxwell does not accept the Galileian transformation. 

For Maxwell, velocity of light is invariant with respect to the reference frame, as it only depends from absolute physical quantities. 

His thought is “absolutely correct”. It is impossible to presume any errors in his theory. 

Nevertheless, such incompatibility towards past schemes is left open. 

Every space has its own “metric”, i.e. a method to define the elementary distance between two points, at discretion, near to each other. 

This is the Euclidean metric of Galileo: 
 

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Einstein’s first synthesis – “spacetime” an assumption of Special Relativity

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This is the cultural scenario in which Einstein (1905) succeeded to propose his first synthesis. 

It is clear, from Maxwell’s theories, that the velocity of light could be an invariant. 

A intense revision of the facts was required to create a new vision, that could explain Maxwell and accept Newton, as a limit case. 

Einstein proceeded through “mental experiments”. 

Let us consider one of these experiments: the one of the rocket at rest and of the rocket moving following an uniform rectilinear motion: 

Assume the rocket contains two light signals: A and B and an observer O, equally distant from both A and B. 

We know that O receives a light signal simultaneously from both A and B. 

We assume that the velocity of light is invariant with respect to the reference frame.
 
 
 
 
 
 
 
 
 
 
 

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Let us now observe a rocket at rest in a reference frame, jointed to it. 

The question is:  do A and B light up simultaneously? Yes, they do. 
 
On the contrary, what happens when we observe a rocket accelerating upward from ground level ...
 
 
 
 
 

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O must perceive the same reality: receiving the light signals from both A and B simultaneously. As light always moves with velocity "c", it becomes evident that in the new reference frame, the two events are no more simultaneous. A emits its signal earlier than the one from B. 
 
 
 
 

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 Einstein’s first synthesis – “spacetime” an assumption of Special Relativity

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Thus, moving over to another inertial reference frame, the notion of time changes.  

In a generic inertial transformation of co-ordinates: x, y, z shift over to new co-ordinates x’, y’, z’. We also know that the notion of time changes. Time is no more an absolute time, but, is a co-ordinate that varies when its reference changes. This is the four dimensional spacetime x, y, z, t.  

It is, necessary now, to find a transformation law that may confirm such a reality. It will be the Lorentz transformation with its corresponding “metric”:  
 

            Galileo’s metric and Lorentz metric 

The velocity of light “c” in vacuum is invariant and corresponds to Maxwell’s equations.  

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Special Relativity – “spacetime diagrams”

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Restrict attention to two spacetime dimensions: the temporal one “ct” and the spatial one, e.g.: the “x” axis 
A point P, in spacetime, is called “event”. It will be located by four co-ordinates in an arbitrary inertial reference frame : 
 

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Special Relativity – “spacetime diagrams”   

The slope of a straight line in this diagram is:                   m = ct / x = c • ( t / x) = c / Vx 

if Vx = c then m = 1, therefore a photon travelling at velocity “c” will describe a straight line in the spacetime diagram at 45°. 

A static particle will be described by a vertical straight line (x does not change; time does ). 

« Nobody would ever think of confusing the relationships between lengths of a Mercator map of the world with the relationships between true distances on the surface of the Earth ». A Mercator map is a projection of the geometry of the globe on a sheet of paper, which has a different geometry… Similarly, a spacetime diagram is a projection of a two dimensional section of spacetime with a geometry  

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on a sheet of paper with a geometry                distances on the page displaying a spacetime diagram must not be mixed up with the true distances in spacetime!». (1) 

(1) James B. Hartle - "Gravity - An introduction to Einstein's General Relativity" - Addison Wesley
 
 
 
 

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Special Relativity – “spacetime diagrams” 

The following figure illustrates these assertions which are not exactly intuitive. Computation makes use of Lorentz transformations: 

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Special Relativity – “co-ordinate transformation” 

Consider a reference frame with uniform velocity with respect to the primary one. It turns out, that in the case of an uniform motion, taking place only along the x-axis, the transformation leads to a diagram having the same origin, but with axes rotated 

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Special Relativity – “rod contraction”

Consider the length Lx in the principal frame (rest length of the rod) and the corresponding length L in the reference frame moving with uniform motion with respect to “x” L. We immediately see they are different, L may be measured starting from Lx and from co-ordinate   using Lorentz transformation ( not the Pythagorean theorem ) 
 

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Special Relativity – Newton’s inconsistency 

« With the success of the Special theory of Relativity, it became apparent that the Newtonian theory of gravity, which had been so successfully applied to the mechanics of the Solar System for almost 300 years, could no longer be exactly correct. 

The Newtonian gravitational interaction is instantaneous. 

The gravitational force on a mass m₁, due to a second mass m₂, is given in magnitude by the following formula: 
 

where r₁(t) and r₂(t) are the positions of the masses at the same instant of time. But in Special Relativity the notion of simultaneity is different in different inertial frames. The Newtonian law could be true in only one frame, and it would then single out that frame from all others. 

The Newtonian law of gravity is thus inconsistent with the principle of relativity ». (2) 

(2) James B. Hartle - "Gravity - An introduction to Einstein's General Relativity" - Addison Wesley 
 
 
 
 
 
 
 
 
 
 

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General Relativity – Equality of Gravitational Mass and Inertial Mass

All General Relativity (GR) is based on this equality. 

Inertial mass tends to oppose the action of a force:   
 

Gravitational mass enters the law of mass attraction:
 

Using different methods, scientists devoted plenty of their time to measure this equality. It was thoroughly checked with the use of precision instrumentation. It is, therefore, a fact accurately consolidated.  
 
 
 
 
 
 
 
 

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General Relativity – The Principle of Equivalence -  ( courtesy of Franco Malerba )   

The equivalence between Gravitational mass and Inertial mass allows to experiment, while in orbit, a situation of “zero gravity”. In the space-shuttle, bodies are in a state of rest. If pushed, they proceed with a rectilinear uniform motion. 
 
 
 
 
 
 
 
 
 
 
 

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General Relativity – The Equivalence Principle 

Earth’s gravitational field is not really uniform, but has a spherical radial symmetry. Moving away from the centre of the planet, it tends to diminish. 

Therefore, it is not strictly true that anywhere, in the space-shuttle, gravity is zero. 

But, in “small” spaces, equality is sufficiently valid with respect to the variation of the field. 

Therefore, its definition, “practically”, sounds as follows: 

«Experiments in a sufficiently small freely falling laboratory, over a sufficiently short time, give results that are indistinguishable from those of the same experiments in an inertial frame in empty space» (3) 

(3) James B. Hartle - "Gravity - An introduction to Einstein's General Relativity" - Addison Wesley
 
 
 
 
 
 
 
 
 
 
 
 

General Relativity – Clocks in a gravitational field 

Assume two observers, both having identical clocks, embedded in an Earth’s gravitational field. Clock A emits a signal at rate  and is positioned at a height h. Clock B is positioned on ground level, and in general, will have a rate . Formula is: 
 

General Relativity - GPS – Global Positioning System 

The rate of signals emitted by the satellites of the GPS system must be corrected, with respect to ground clocks, because satellites move at high speed and because their gravitational potential is different from the one of the ordinary user. 

Fractional correction in rate for time dilation due to satellite’s velocity is: 
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Fractional correction in rate for the gravitational potential is: 
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The gravitational correction is bigger than the correction for time dilation.  
 
 
 
 
 
 
 
 
 

General Relativity – Spacetime is curved  

«To better explain this concept, consider an inertial reference frame K and a non inertial reference frame K' rotating in an uniform way with respect to K. Additionally, consider a circumference jointed to K. With respect to K, the ratio between the circumference at rest and its diameter is . With respect to K', rotating anti-clockwise, circumference rotates clockwise. Each small segment of the circumference is seen moving by K' with a certain velocity “v” [radial].

At a certain point each small segment, forming the circumference, shrinks with respect to K', following Lorentz contraction, therefore, the ratio between circumference and diameter, with respect to K', is different from (the diameter does not follow Lorentz contraction, as it does not move lengthwise with respect to K). 
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« ... [so] it is proved that space with respect to an accelerated reference frame is not flat, but curved. The Euclidean geometry no longer applies. Since, a gravitational field is equivalent to an accelerated reference frame, the spacetime is curved by a gravitational field » - E-school - Arrigo Amadori – “Relatività generale (RG) (sintesi introduttiva)”  ” 
 
 
 
 
 
 
 
 
 

General Relativity - Metric 

The Metric of General Relativity (GR) is a generalization of the Restricted Relativity Metric  ... 

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What is subscript must be added to all possible combinations …  the temporal co-ordinate, or the x, or the y, or the z and could also be … therefore, the sum will have 16 addenda, taking into consideration all the possible combinations two by two. 

In  Contains the information of the mass quantity and of its distribution, which, as previously seen, causes the deformation of spacetime and decides its structure. 

It is clear that metric depends from  as its structure depends from the spacetime in question . 
 

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