Definition: Proof by Contradiction (Reductio ad Absurdum)

The logical background of a proof by contradiction is Aristotle’s law of non-contradiction. A rigorous proof by contradiction proof of a theorem follows the standard method of contradiction used in science and mathematics and should be convincing as much as possible. For the first, we assume that a claim/a theorem/a proposition/a statement et cetera which has to be proved, is true. One then proceeds to demonstrate that a conclusion drawn from such a claim/a theorem/a proposition/a statement et cetera leads to a contradiction. Hence, the supposed claim/theorem/proposition/statement et cetera is deemed to be false. Consequently, we are then led to conclude that it was wrong to assume the claim/the theorem/the proposition/the statement was true. Thus far, the claim/the theorem/the proposition/the statement is proved to be false.

Definition: Thought Experiments

Properly constructed (real or) thought experiments (as devices of scientific investigation) can be used for diverse reasons in a variety of areas. Thought experiments can help us to investigate some basic properties of nature even under conditions when it is too difficult or too expensive to run a real experiment. Furthermore, a thought experiment can provide some evidence against or in favor of a theory. However, a thought experiment is not a substitute for a real experiment.

Definition: Newton’s Law of Gravitation from the Standpoint of a Moving Observer O

The influence of Isaac Newton’s masterpiece Philosophiae naturalis principia mathematica on the development of modern science, physics and astronomy is still of historical importance. Isaac Newton (1642-1727) published on 5 July 1686 in his work Philosophiae Naturalis Principia Mathematica on page 198 (Prop. LXXVI. Theor. XXXVI) his well known universal law of gravitation. Newton defines the force of gravity in the following way (Figure 1):

Translated from the Latin into English.

“The attractive force of every point decreases in the duplicate ratio of the distance of the body attracted ; I say, that the whole force with which one of these spheres attracts the other will be reciprocally proportional to the square of the distance of the centres.”

For the first, Newton’s key word is proportional. Newton’s statement of proportionality is the source of Newton’s gravitational constant Big _OG, known to physicists as Big _OG. For the second, Newton himself is of the opinion, that his law of universal gravitation is valid under conditions of inertial frames of reference. Newton’s law of universal gravitation from the standpoint of a co-moving observer O in a special relativity consistent form written as an equation states

where _OF is the force due to gravity between the two masses, _OG is the gravitational constant as measured by a moving observer, _Om₁ is the mass of object 1, _Om₂is the mass of object 2. Thus far, _Od = c ´ _Ot is the distance between the centers of these two masses (_Om₁ and _Om₂) as measured by the moving observer O.

Scholium

Under conditions of inertial frames of reference invariant quantities are the same for all observers (an observerindependent quantity). In special relativity, the total force or the net force (the sum of forces acting on an object) is equal to zero. But this does not mean that there are no forces. In general, force is a vector quantity, which means force has a direction and a magnitude. Sometimes, force is denoted using boldface such as F.

Definition: Newton’s law of gravitation from the standpoint of a stationary observer

Let _RF denote the force under conditions of special relativity as determined by a stationary observer R. Let _Rm₁ denote the “relativistic” mass as measured by a stationary observer R. Let _Rm₂ denote a second “relativistic” mass as measured by a stationary observer R. Let _RG denote the Newtonian gravitational constant as determined by a stationary observer R. Let _Rt denote the time as determined by a stationary observer R. Let c denote the speed of the light in vacuum. Let _Rd = c ´_Rt denote the distance between the centers of the two masses _Rm₁ and _Rm₂ as measured by the stationary observer R. Newton’s law of universal gravitation from the standpoint of a stationary observer R written as an equation states

Scholium.

Under conditions of inertial frames of reference, we expect that a Newtonian gravitational constant which is determined by a stationary observer is equivalent with a Newtonian gravitational constant as measured by a moving observer. Especially, due to Newton’s third law, under conditions of special theory of relativity, we do expect that is _OF = _RF.

Definition: Rest mass _Om and relativistic mass _Rm

The relative motion between two inertial frames of reference, where Newton’s laws are valid, causes observers in those frames of reference to measure different values of the particle’s energy (mass) and time. Especially, the notion “relativistic mass” _Rm of an object and the time is dependent on the relative velocity v of the observer and depends at the end on the observer’s frame of reference. Under circumstances where the relative velocity between two inertial frames of reference is equal to zero, observers will measures the same values for particle’s energy (mass) and time and distances too. Thus far, according to special relativity theory [10] - [12] it is

Scholium

Einstein special theory relativity defines mass at least in two different ways. The “rest mass” or “invariant mass” denoted as _Om is an invariant quantity which is measured by an observer at rest relative to _Om has the same value for all observers in all reference frames. The “rest mass” _Om of an object is identical with the Newtonian mass as measured by an observer moving along with the object, while a rest relative to the moving object. The “relativistic mass” _Rm is dependent on the velocity of the observer. The term “relativistic mass” _Rm was first defined by Gilbert N. Lewis and Richard C. Tolman in 1909 [13] .

Definition: The time of a stationary observer _Rt and the time of a co-moving observer _Ot

Time is dependent on the observer’s reference frame. Especially, clocks moving at close to the speed of light c will slow down with respect to a stationary observer R (observer at rest). Thus far, let _Rt denote the time as measured by a stationary observer, i.e. the relativistic time. Let _Ot denote the time as measured by a moving observer. The relationship between the time _Ot as measured by a clock moving at constant velocity v in relation to the time _Rt as measured by a clock of a stationary observer is determined by Einstein’s relativistic time dilation [14] as

where _Ot denotes time as measured by a moving observer, _Rt denotes the time as measured by a stationary observer, v denotes the relative velocity and c denotes the speed of light in vacuum. Equally, it is

or

Scholium

Coordinate systems can be chosen freely, deepening upon circumstances. In many coordinate systems, an event can be specified by one time coordinate and three spatial coordinates. The time as specified by the time coordinate is denoted as coordinate time. Coordinate time is distinguished from proper time. The concept of proper time, introduced by Hermann Minkowski in 1908 and denoted as _Ot, incorporates Einstein’s time dilation effect. In principle, Einstein is defining time exclusively for every place where a watch measuring this time is located.

“... Definition ... der ... Zeit ... für den Ort, anwelchemsich die Uhr … befindet ...” [15] .

In general, a watch is treated as being at rest relative to the place where the same watch is located.

“Eswerdefernermittels der imruhenden System befindlichen ruhend en Uhren die Zeit t [_Rt, author] des ruhenden Systems ... bestimmt, ebensowerde die Zeit τ [_Ot, author] des bewegten Systems, in welchensichrelativzuletzteremruhendeUhrenbefinden, bestimmt...” [16] .

Only, the place where a watch at rest is located can move together with the watch itself. Therefore, due to Einstein, it is necessary to distinguish between clocks as such which are qualified to mark the time _Rt when at rest relatively to the stationary system R, and the time _Ot when at rest relatively to the moving system O.

“Wirdenkenunsfernereine der Uhren, welcherelativzumruhenden System ruhenddieZeit t [_Rt, author], relativzumbewegten System ruhend die Zeit τ [_Ot, author] anzugebenbefähigtsind ...” [17] .

In English:

“Further, we imagine one of the clocks which are qualified to mark the time t [_Rt, author] when at rest relatively to the stationary system, and the time τ [_Ot, author] when at rest relatively to the moving system….”.

In other words, we have to take into account that both observers have at least one point in common. The stationary observer R and the moving observer O are at rest, but at rest relative to what? The stationary observer R is at rest relative to a stationary co-ordinate system R, the moving observer O is at rest relative to a moving co- ordinate system O. Both co-ordinate systems can but must not be at rest relative to each other. The time _Rt of the stationary system R is determined by clocks which are at rest relatively to that stationary system R. Similarly, the time _Ot of the moving system O is determined by clocks which are at rest relatively to that the moving system O. What is the time marked by the clock when viewed from the stationary system? What is the time marked by the clock when viewed from the moving system? In last consequence, due to Einstein’s theory of special relativity, a moving clock (_Ot) will measure a smaller elapsed time between two events than that measured by a non-moving (inertial) clock (_Rt) between the same two events.

Definition: The normalized relativistic time dilation

As defined above, due to Einstein’s special relativity, it is

The normalized relativistic time dilation relation [18] follows as

Definition: The distance _Od and the distance _Rd

In general, it is distance = speed ´ time. The time as such depends on the frame of reference in which it is measured or in other words a moving clock will more slowly. In general, it follows that

where _Od denotes the distance as measured by a moving observer, _Ot = _Ot₂ − _Ot₁ denotes the (period of) time as measured by a moving observer, _Rd denotes the distance as measured by a stationary observer, _Rt = _Rt₂ − _Rt₁ denotes the (period of) time as measured by a stationary observer, v denotes the relative velocity between the moving O and the stationary R observer and c denotes the speed of light in vacuum.

Definition: Einstein’s field equation

Newton’s theory of gravity has been superseded by Einstein’s (1879-1955) new theory of gravity but continues to be used as an approximation of the effects of gravitation. Leaving aside that Newton’s theory of gravity is under dispute, Newton’s gravitational constant is still of use and part even of Einstein’s general relativity theory [19] too. Einstein’s field equations are determined as

where _RG denotes Newton gravitational constant. Einstein’s general relativity theory does seem to be the correct theory of gravitation at least at low energies. As can be seen, curvature of space-time is proportional to Newton’s constant of gravitation.

Definition: The relativistic Doppler effect

Redshift can be measured by determining the wavelength of a known transition, such as hydrogen a-lines for distant quasars, and finding the fractional shift compared to a stationary reference. The light from the stars or from distant galaxies has distinct spectral features i.e. due to the characteristics of the atoms in the gases around the stars. These spectra can be examined, they are found to be shifted (Doppler shift). The measured (red) shifts are usually stated in terms of a so called z parameter. The measured Doppler red shifts can be used to calculate the recession velocity v of stars or galaxies, presuming that the Hubble law is valid. We obtain

A z value of 8.68 was measured by Zitrin et al. [20] .

The following theory is based on the axiom:

Claim (Theorem. Proposition. Statement)

Under conditions of special relativity (inertial frames of reference) Newton’s gravitational constant is not a constant. In principle, a co-moving observer O and a stationary observer R will not agree on the value of Newton’s gravitational constant. In general, we must accept that

Proof.

In general, axiom I is determined as

Multiplying this equation by the “rest mass” _Om₁, we obtain the mass as determined by co-moving observer as

According to Einstein’s theory of special relativity the laws of physics are the same in any inertial frame of reference. A stationary observer R will ascribe the value _Rm₁ to the same object. Due to special relativity we obtain

Multiplying this equation by _Rm₂, we obtain

Two different observers moving with constant velocity relative to each other will have a different view on the product of two masses. Due to special relativity, the equation before can be rearranged as

Quod erat demonstrandum

Scholium

Two different observers moving with constant velocity relative to each other will obtain the same value of the product of the two masses i.e. equation by. This simple equation does not change everything we thought we understood about Newton’s gravitational constant Big G. But the same equation makes it evident that under conditions of special theory of relativity mass as such has no influence on a possible variation of Newton’s gravitational constant Big G.

Claim (Theorem. Proposition. Statement)

Under conditions of the special theory of relativity, we must accept that

Proof

In general, it is

Multiplying this equation by the “rest mass” _Om₁ and by the “rest mass” _Om₂ we obtain

Dividing this equation by _OG/_Od² it is

Due to special relativity, this equation can be rearranged as

or as

In general, the equation simplifies as

Quod erat demonstrandum.

Claim (Theorem. Proposition. Statement)

Under conditions of the special theory of relativity, where the relative velocity between the mass _Om₁ and the mass _Om₂ is equal to v = 0 while moving at constant relative velocity v with respect to a stationary observer R, we must accept that

Proof

In general, it is

Multiplying this equation by Newton’s gravitational constant _OG we obtain

At this point, our assumption is that Newton’s gravitational constant as determined by a stationary observer R and denoted by _RG is identical with Newton’s gravitational constant _OG as determined by the moving observer O. In general it is assumed, that Newton’s gravitational constant is independent of the frame of reference. Thus far, we change the equality above and do obtain

Multiplying this equation by (_Om₁ ´ _Om₂)/_Od² it follows that

Due to special relativity, we rearrange this equation as

This equation simplifies as

Quod erat demonstrandum.

Scholium.

Let J denote the co-moving observer. Let L denote the stationary observer. Let Œ denote mass 1, let  denote mass 2. The relative velocity between the moving observer J and the mass Œ and the mass  is equal to v = 0. In other words, the moving observer J is at rest relative the mass Œ and the mass . Mass Œ and mass  are moving at the same time with a constant relative velocity with respect to the stationary observer L. Table 1 may illustrate this experimental situation in more detail.

The moving observer J is at rest relative to the mass 1 denoted as Œ and at the same time at rest relative to the mass 2 denoted as . The moving observer J ascribes to the mass Œ the value _Om₁ and at the same time to the mass  the value _Om₂. Within the frame of reference of the moving observer J, the relative velocity between the two masses is zero. At the same time, from the standpoint of the stationary observer L, both masses are moving with a certain relative and constant velocity v relative to the frame of reference of the stationary observer L. Therefore, the stationary observer L ascribes to the mass Œ the value _Rm₁ and to the mass  the value _Rm₂. Written as a mathematical formula, we obtain

Under these experimental conditions, it appears to be justified to conclude that Newton’s gravitational constant Big _OG is a constant [21] .

Claim (Theorem. Proposition. Statement)

Under conditions of special theory of relativity it is

Proof

In general, it is

Newton’s law of universal gravitation follows as

Consequently, due to Theorem 3.3, it is

or according to special relativity

This equation can be simplified as

which is equal to

Thus far, we obtain the relationship

Quod erat demonstrandum

Scholium

Let Œ denote mass 1, let  denote mass 2. Let J denote the co-moving observer. Let L denote the stationary observer. The moving observer J is at rest relative _Om₁. The stationary observer L is at rest relative _Om₂. The following 2 × 2 Table 2 may illustrate the relationships above.

Due to the constant relative velocity, which is different from zero, the moving observer J will ascribe the mass _Rm₂ to the mass . At the same time, the stationary observer L ascribes the mass _Rm₁ to the mass Œ. This has no influence on the product of both masses. We obtain

The product of the two masses m₁ and m₂ as calculated by the moving observer is equivalent to the product of the two masses m₁ and m₂ as calculated by the stationary observer.

Claim (Theorem. Proposition. Statement)

Under conditions of special theory of relativity, it is

Proof

In general, it is

Newton’s law of universal gravitation follows as

Consequently, due to Theorem 3.3, it is

or according to special relativity

This equation can be simplified as

which is equal to

Thus far, we obtain the relationship

According to the Theorem 3.2, we obtain equally

In general, it is

Rearranging this equation, it follows from the above that

We simplify this equation as

or as

In general, we obtain

Quod erat demonstrandum.

Scholium

The moving observer O and the stationary observer R will agree on the value on Newton’s gravitational constant Big G only if v = 0, otherwise not. The value of Newton’s gravitational constant Big G is reference frame dependent, Newton’s gravitational constant Big G is not a constant.

Claim (Theorem. Proposition. Statement)

An international team of astrophysicists [20] discovered a distant galaxy, called EGSY8p7, with a spectroscopic redshift of z = 8.68 by observing its characteristic hydrogen signature. This red shift is equivalent to a recessional velocity of the distant galaxy EGSY8p7 as v = 0.9788812110358344´ c. Assuming that v is approximately identical with the relative velocity between the galaxy EGSY8p7 and our earth, Newton’s constant Big G of the distant galaxy EGSY8p7 _oG can be calculated approximately as

Proof

Due to our theorem before, it is

The relative velocity of the distant galaxy is approximately v = 0.9788812110358344´ c as calculated in this paper while Newtons’s constant _RG is about ~6.674 × 10^-11 [N×m²/kg]. We obtain

We obtain

Quod erat demonstrandum.