The Generalised Mass-Energy Equation E = Ac2M; its applications in Cosmology

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A Plausible Model for Pre-Big Bang Cosmology, Gamma Ray Bursts, Quasars etc.
Ajay Sharma

Community Science Centre. DOE. Post Box 107. Shimla 171001 HP INDIA
Alternate Email physicsajay@lycos.co.uk

PACS 98.80.–k, 98.54.Aj, 98.80.Cq, 97.60.Bw

Abstract
Newly suggested mass-energy inter convertibility equation D E = Ac²D M implies that energy emitted in any form in any reaction on annihilation of mass (or vice- versa) can be equal, less and more than predicted by D E = D mc² (precisely confirmed in nuclear reactions). The nature and variation of conversion coefficient A is precisely same as other constants or co-efficients of proportionally in existing physics. It successfully explains the energy emitted (10⁴⁵J) in Gamma Ray Bursts (duration 0.1s-100s) with value of A i.e. 2.57´ 10¹⁸. Also energy emitted by quasars and supernovas etc. can be explained with higher value of A. Recent work at SLAC confirmed discovery of a new particle Ds (2317), whose mass is far less than current estimates, the same can be explained with help of equation D E = Ac²D M with value of A more than one. D E = Ac²DM, is the first equation which mathematically explains that mass of universe 10⁵⁵kg was created from dwindling amount of energy (10^-4444J or less) with value of A equal to 2.568´ 10^-4471 J or less. Whereas D E = D mc² predicts the mass of universe 10⁵⁵kg was originated from energy 9´ 10⁷¹ J. Hence in nature in various processes the conversion of mass to energy is not always consistent with D E = D mc².

1.0 The generalised mass-energy inter convertibility equation D E = Ac²DM
The law of conservation of mass or energy existed in literature since 18^th century (or may be even before informally) the French chemist Antoine Lavoisier was the first to formulate such a law for chemical reactions. The very first idea of mass-energy inter conversion was given by Fritz Hasenohrl [1]. Einstein [2] derived first inter conversion-equation between light energy (D L) and mass D L = c² D m, and speculated from it the general equation (for every type of energy) D E = D mc² ( precisely confirmed in nuclear reactions). To explain the astounding amount of energy emitted especially by some heavenly bodies and possibly visualise the origin of universe before Big Bang, the equation is D E = Ac²D M is anticipated or visualised by author [3-4].
Here the derivation involves calculation of infinitesimally small amount of energy dE when small amount of mass dm is converted (in any process) into energy. The energy may be in any form i.e. light energy, sound energy, energy in form of invisible radiations etc or energy may co-exist in various forms as in case of atom bomb then

dE µ dm

In the existing literature in nuclear reactions conversion factor c² between mass and energy has been experimentally confirmed. Thus in above proportionality, it can be taken in account as,

dE µ c² dm or dE = Ac² dm (1)

where A is co-efficient of proportionality, and is dimensionless variable. Its nature and status are precisely same as those of other constants or co-efficients of proportionally in existing physics. Let in some conversion process mass decreases from M_i to M_f and energy increases from E_i to E_f . Initially when no mass is converted into energy, E_i = 0. Thus integrating Eq. (1) we get,

E_f - E_i= Ac² (M_f – M_i) (2)

D E = Ac² D M or Energy evolved = Ac²(decrease in mass) (3)

The general mass energy inter convertibility equation D E = Ac² D M is not in confrontation with D E = D mc² , but is applicable in chemical reactions, nuclear reactions and reactions taking place in heavenly bodies depending upon situations with different values of A. Thus D E = Ac² D M is general formulation and D E = D mc² is its special case.

If value of A =1, then Eq.(3) is simply D E = D mc², if A >1 then energy emitted more than D E = D mc² and if A<1 then energy emitted is less than D E = D mc² . In D E = D mc² , c² is universal constant for all types of existing and expected reactions reaction Whereas in equation D E = Ac² D M, conversion factor is Ac², rather than c . Thus the generalised equation may be stated as

"The mass can be converted into energy or vice-versa under some characteristic conditions of the process, but conversion factor may or may not always be c² (9 ´ 10¹⁶ m²/s²) or c^-2"

The Eq. (3) can be obtained by method of dimensions.
Let the energy emitted (D E) , depends upon annihilated mass (D M) as dimensions a, depends upon speed of light c, as dimensions b and depends upon time t as dimensions c. Thus

D E µ (D M)^a c^bt^c or D E = A (D M)^a c^bt^c (4)

where A is constant of proportionality and is called Conversion Co-efficient. Hence

ML²T^-2 = A M^a (LT^-1)^b T^c = A M^a L^bT^–b+c

or a=1, b=2 and –2 = –2 +c or c =0

Thus, D E = A D Mc²t^o = Ac²DM(3)

1.2 The variation in magnitude of ‘A’ is consistent with existing Physics.
Now obvious question is how should the co-efficient A vary i.e. on what factors does it depend or get influenced? The answer for question of dependence or variation of co-efficient of proportionality A is precisely same as answer for all other proportionality constants or co-efficients in existing physics. The co-efficient A does not have any special characteristics neither in regard to its origin nor interpretation and estimation. All such constants or co-efficients of proportionality in existing physics depend upon the intrinsic characteristics conditions and parameters which influence the results directly or indirectly; hence the same is precisely true for A . The constant of proportionality may arise by method of conceptual derivation or by method of dimensions always determined experimentally. The co-efficient A is dimensionless due to reason that it is introduced in existing equation of energy, and dimensions of energy has to be ML²T^-2 same in both sides, in F=kma , k is also dimensionless. In physics the same entity may behave in different ways under different conditions. For example a single wave of radiations behaves like both wave and particle; also atomic particle electron behaves like both wave and particle depending upon characteristic conditions. Thus status of conversion co-efficient A and its magnitude is consistent with existing physics.

If the constant of proportionality varies from one system to other (which is realistic situation in many cases), then it is termed as co-efficient e.g. coefficient of viscosity, co-efficient of thermal conductivity, co-efficient of elasticity (Young’s modulus, Bulk Modulus, Modulus of rigidity) etc. Analogously Hubble’s constant must be better called Hubble’s co-efficient, as there is significant variation in its value for various heavenly bodies. The generalised trends of various constants or co-efficients of proportionality are shown in Table I.

Table I

Sr No	Constant or co-efficient of proportionality	Variation in magnitude
1	Hubble’s constant	50 to 80 kilometres per second-Mega parsec (Mpc)
2	Co-efficient of thermal conductivity	0.02Wm^-1K^-1 to 400 Wm^-1K^-1
3	Coefficient of elasticity	(3-200) ´ 10⁹ N/m²
4	Co-efficient of viscosity	1.05´ 10^-3 poise to 19.2´ 10^-6 poise
5	Decay constant ( 0.693/T_1/2)	10¹⁵s^-1 – 10 ^-10 s^-1 ( general trend)
6	Constant in Second law of motion ( F=kma )	k=1
7	Universal Gravitational constant G.	6.673(10) × 10^-11 m³ kg^-1 s^-1 (showing increase )
8	Acceleration due to gravity g	9.80665 m s^-2 ( varies from place to place)
9	Einstein’s conversion constant (ΔE = Δmc² )	c² or 9 × 10¹⁶ m²s^-2(universal constant)
10	Generalized equation’s ( ΔE = Ac²ΔM) conversion co-efficient	Ac² or A 9 × 10¹⁶ m²s^-2

The co-efficient of thermal conductivity, K is given by

K = Qd / A(T₁-T₂)t (4)

where Q is heat transmitted, d is thickness between surfaces, t is time for which heat flows, T₁ is temperature of one face and T₂that of the other. Similarly Hubble constant, H is ratio of velocity of recession, V and distance of heavenly body, D i.e.

H = V / D (5)

Thus to determine H, V and D both are measured. The decay constant in radioactivity,

λ = 0.693/T_1/2(6)

The value of T_1/2( half life time) elementary particles vary from 10^-6s to 10^-23s and for uranium-238 is 4.5 billion years, depending upon their inherent characteristics and accordingly decay constants vary. Further decay constant cannot predict why half life of one particular particle is 10^-10s and other element 1 billion year. It simply equates physical quantities in LHS and RHS. Similar is the status of other proportionality constants or co-efficients and including A as in Eq.(3)

Even in Einstein’s ΔE = Δmc², the conversion constant between mass annihilated and energy created (in any form) is c² (ΔE/Δm), which is like universal constant ( as k in F = kma) . However, there are proposals for increase or decrease in value of c [5-6]. If these proposals for variations in values of c matured then Einstein’s equation ΔE = Δmc²will become quantitatively invalid. Then equation ΔE = Ac²ΔM will be applicable, as it predicts energy emitted can be less, equal or more than ΔE = Δmc² due to presence of A. Similarly the value of A is given by

A = ΔE/c²ΔM

Thus irrespective of status of c the generalized equation remains valid.

The variation of value of A can be understood in three categories, as emission of energy is observed in different reactions e.g. chemical reactions, nuclear reactions and reactions taking place in heavenly bodies. Depending upon the inherent characteristics of the reaction the energy emitted in each type of reaction is different, thus like other proportionality factors (constants or co-efficients) the value of A varies as described below.

(i) ) If energy emitted (ΔE) corresponding to annihilation of mass (ΔM ) is such that ratio (ΔM /ΔE ) is equal to 1/c², then in generalized equation ΔE = Ac²ΔM, the value of A equals unity (A =1)

ΔE /ΔM depends upon inherent characteristics of the process. In generalized mass energy inter convertibility equation ΔE = Ac²ΔM, the value of A is unity (A =1) for nuclear reactions. Thus in this case the generalized equation reduces to Einstein’s ΔE = Δm c². ΔE = Δm c², is a basic or standard equation in nuclear physics as used in deriving relationship 1amu = 931.49 MeV, hence all masses are expressed using it. Here basic assumption is that speed of light will always remain constant, i.e.

c² = ΔE/Δm = Any type of energy created / Mass annihilated.

otherwise all estimations will vary. Hence this discussion gives another but indirect method of determination of speed of light. Also there are both theoretical and experimental variations in value of c [5-6], as fine structure constant (= e²/c) is reported to be increasing over cosmological timescales, implying slowing down of speed of light, c

(ii) If energy emitted (ΔE) corresponding to annihilation of mass (ΔM) is such that ratio (ΔM /ΔE) is less than 1/c², then in generalized equation ΔE = Ac²ΔM, the value of A is more than unity (A >1).
(a) In this case practical example is energy measured in case of Gamma Ray Bursts and Quasars. It is inherent characteristic of these heavenly events that energy emitted for mass is far higher than predicted by ΔE = c²ΔM
(b) In SLAC mass of particle, Ds (2317) experimentally observed, was found less than expected estimates [7] , it can also be explained with generalized mass –energy inter convertibility equation, ΔE = Ac²ΔM with value of A more than one.

(iii) ) If energy emitted (ΔE) corresponding to annihilation of mass (ΔM ) is such that ratio (ΔM /ΔE ) is more than 1/c², then in equation ΔE = Ac²ΔM, the value of A is less than unity (A <1)
In generalized equation ΔE = Ac²ΔM, the value of A is less than unity (A <1) if small energy (ΔE ) is materialized to large mass (ΔM ).
(a) Before Big Bang mass of the order of 10⁵⁵kg has been produced from diminishing amount of energy which may be regarded as present in the space at that time, and value of A is less than one. In this case Einstein’s ΔE = Δmc², requires reserve energy of order of 10⁷² J created out of nothing this energy is materialized to mass 10⁵⁵kg (E/c²).
(b) In chemical reactions ΔE = Δmc² has not been confirmed, but regarded as precisely true which is unscientific as there is vast difference between chemical and nuclear reactions. The reason for non-confirmation is that experimental precision is too less to measure the mass annihilated in the process. As experimental precision is increasing, and at some stage experiments are conducted and amount of energy created (ΔE ) is found less than predicted by ΔE = Δmc² corresponding to mass annihilated (ΔM ) ; then value of A less than one may be confirmed.

2.0 D E = Ac² D M in Cosmology
For determination of A, the value of D M i.e. mass annihilated in case of heavenly body is required; which can not be directly measured like many other parameters. Thus, initially for simplicity or calibration (standard or reference can be chosen) the magnitude of value of D M is regarded as 4.322´ 10⁹kg i.e. mass annihilated in case of sun (luminosity of the sun is 3.89´ 10²⁶ Js^-1), thus

D M = D E/c² = 3.89´ 10²⁶ Js^-1/ 9´ 10¹⁶ m²s^-2 = 4.322´ 10⁹kg (7)

If for some cases the value of D M is experimentally measured then its actual value (D M ) can be used instead of Eq.(7).

2.1 Gamma Ray Bursts
Gamma ray bursts (GRBs) are intense and short (approximately 0.1-100 seconds long) bursts of gamma-ray radiation that occur all over the sky approximately once per day and originate at very distant galaxies (several billion light years away). GRBs are the most energetic events after the Big Bang in the universe and energy emitted is approximately 10⁴⁵J with the most extreme bursts releasing up to 10⁴⁷ J. This energy cannot be explained with D E = c^2Dm (precisely confirmed in nuclear reactions). This is also the amount of energy released by 1000 stars like the Sun over their entire lifetime! It implies that for annihilation of dwindling mass in short time unimaginably high amount of energy is emitted, which can be explained with help of D E = Ac^2DM with exceptionally high value of A. If for simplicity the value of D M can be taken standard as in Eq.(7) as actual estimate of D M for GRBs is not available, then

A_grb = D E / c² D M = 10⁴⁵/ 9´ 10^16´(4.32´ 10⁹) = 2.57´ 10¹⁸ (8)

or D E =2.57´ 10¹⁸ c² D M (9)

Hence all conversions of mass to energy in nature, is not always according to D E = c^2Dm. , where c is the conversion factor like universal constant. In the GRBs intense and short bursts of gamma-ray radiation are emitted; which implies for small mass (simply gamma rays), in small region, in small time huge amount of energy is liberated. It is direct confirmation for D E = Ac^2DM with very high value of A i.e. for annihilation of small mass (burst of Gamma Ray), in short time enormous amount of energy is emitted (in this case 2.31´ 10³²J for annihilation of 10^-3kg) which is 2.57´ 10¹⁸ times more than D E = c^2Dm. However the actual value of A_grb will be more when exact values of D m corresponding to energy emitted will be experimentally determined, instead of standard value as given by Eq.(7).

2.2 Quasars
The observations taken with the 2.5-meter Isaac Newton Telescope at La Palma in the Canary Islands reveals that the quasar is 4 million-billion ( 15.56´ 10⁴¹ Js^-1) to 5 million-billion times brighter than the Sun or this energy is thousand times more than emitted by the brightest galaxy. The most peculiar characteristics of Quasar is reported by Arav et al. [8] that this prodigious amount of energy is generated in a small region approximately one light year across. By comparison the diameter of the Milky Way is about 100, 000 light years. It implies corresponding to a small region (a measure of mass and its hence annihilation) mammoth amount of energy is emitted in case of Quasars. D E = Ac^2DM is useful in explaining such aspects. Now

A_qu = D E / c² D M = 15.56´ 10⁴¹Js^-1/9´ 10^16´(4.32´ 10⁹) = 4´ 10¹⁶

With this value of the generalized mass-energy inter convertibility equation becomes,

D E =4´ 10¹⁶ c² D M (10)

Thus corresponding to small mass (size) energy emitted is more thus comparatively smaller quasars or in general smaller bright objects are feasible. So in small region even when small amount of mass is annihilated, huge amount of energy is emitted. The lower limit of Quasars mass is not yet determined, Vestergaard [9]. It is further justified from the fact that the Quasars possibly or inexorably ending as super massive black holes, presently the maximum mass is of the order of 2´ 10⁴⁰ kg, Vestergaard [10]. Thus inspite of emitting huge amount of energy in own life time, significant amount of matter is remnant in Quasar and which are expected to behave like super massive black hole, this aspect is easily explained on the basis of D E = Ac^2DM, with high value of conversion co-efficient, A. Normally a black hole have density of the order of 10¹⁸kg/m³, and even light cannot escape from it, may be regarded as formed after numerous cycles.

It can be concluded that to attain such state Quasars must under go series of large number of exceptionally intense compressions utilizing energy produced in itself. But energy used for this purpose (internal changes) is not taken in account in current measurements of luminous energy, implying that total energy (including measurable and immeasurable) is far higher than current estimates i.e. A_qu may be more than 4´ 10¹⁶ ( it is only for luminous energy). This large amount of energy emitted by Quasar and other heavenly bodies is consistent with D E = Ac²DM with higher values of A. Similarly energy emitted by supernova and other bodies can be explained. Thus according to this equation D E = Ac²D M more energetic and abundant such explosions in universe are feasible and universe is more long lived compared to predictions of D E = D mc² as for smaller mass huge amount of energy is emitted. The values of A for various heavenly bodies are shown in Table II.

Table II : The values of Conversion–Coefficients (A) for various heavenly bodies and phenomena.

Sr. No	Event emitting energy	Energy (Joules)	D M (kg)	A=D E / c² D M
1	Sun	3.89´ 10²⁶	4.32´ 10⁹	1
2	Gamma Ray Burst	10⁴⁵	4.32´ 10⁹	2.57´ 10¹⁸
3	Quasar	15.56´ 10⁴¹	4.32´ 10⁹	4´ 10¹⁶
4	Supernova	5´ 10³⁵	4.32´ 10⁹	1.286´ 10⁹
5	Bright Star	2.73 ´ 10³¹	4.32´ 10⁹	7.02 ´ 10⁴
6	Creation of mass of universe before big bang (10⁵⁵kg)	10^-4444	4.32´ 10⁹	2.568´ 10^-4471

3.0 Creation of mass of universe (10⁵⁵kg) before Big Bang
The Big Bang Theory assumes that initially (t=0) whole mass 10⁵⁵kg of universe was infinitely compact and in singular state enclosing a space even smaller than an atomic particle instantaneously exploded in gigantic detonation ( various heavenly bodies figured ) and ever since the universe is expanding, Hawking [11]. How the whole mass of universe was formed and condensed to infinitely compact point? How explosion was triggered causing expansion, reduction in temperature and density drastically? Which source provided energy for these events? Why universe of mass 10⁵⁵kg, instead of getting into a point mass of density of undreamt magnitude did not start moving away in the beginning itself? Like this that energy would have been saved which was consumed in making universe a point mass and causing explosion. Thus Big Bang theory assumes excess energy in the universe.

Currently, transformation of mass to energy or vice-versa is explained with D E = D mc² i.e. a gamma ray photon of energy at least 1.02 MeV ( 1.623´ 10^-13 J ) gives rise to electron and positron pair (18.2´ 10^-27 kg) is consistent with it. The mass of universe is estimated to be nearly 10⁵⁵ kg, thus as above it must have been materialized from energy (D E=D mc²) i.e. 9´ 10⁷¹J. Further additional energy ( which may be infinitely large i.e. unimaginably high to be appraised) is required to change mass 10⁵⁵kg into a point of exceedingly high density, and raise the temperature, trigger an explosion and to impart kinetic energy to it (even now accelerating outward continuously). Now it has to be assumed that energy 9´ 10⁷¹J and spectacular amount of additional energy (may be infinitely large amount of energy for above events) as mentioned above is created from nothing or naught or cipher automatically and spontaneously. The law of conservation of energy does not permit creation of mass out of nothing at all (further on such highest scale), hence the law was not obeyed at that stage according to D E=D mc². How the energy of the order of 9´ 10⁷¹J was produced? How the energy materialized to mass (gamma ray only changes into electron –positron pair when passes near the filed of nucleus) ? Thus conversion of energy to mass is conditional. All these intrigues are neither answered by detractors nor adherents of Big Bang Theory, and are open for plausible elucidation.

The general mass-energy inter convertibility equation D E = Ac² D M predicts that in this primordial bang (exceptionally-2 super special event), diminishingly small pulse of energy, say 10^-4444 J ( or less) equivalent to 2.4´ 10^-4443calorie ( or less), can manifest itself in mass 10⁵⁵kg if the value of A is regarded 2.568´ 10^-4471 . The energy 10^-4444 J or less is regarded as to exist inherently in the universe, even when there was no material particle or when process of formation of space started.

The primordial value of conversion coefficient A_uni :

Now the value of various parameters can be written as

A_uni = 10^-4444/9´ 10^16´4.32´ 10⁹ = 2.568´ 10^-4471

or D E =Ac² D M = 2.568´ 10^-4471 c² D M (11)

p Thus D E = Ac²D M, is the first equation which at least theoretically predicts that universe (10⁵⁵kg) has been created from minuscule or immeasurably small amount of energy (10^-4444J or less, which may be easily available compared to 9´ 10⁷¹ J ).Whereas D E=D mc²predicts that mass of universe ( 10⁵⁵kg) has originated from mammoth energy i.e. 9´ 10⁷¹ J ( plus additional energy as cited above ). Thus the generalised equation explains the origin of mass of universe with ease and simplicity; and in addition universe is more long lived than present estimates. Thus inter convertibility of energy to mass was there, but for small energy amount of mass created was much higher than D E =D mc²

4.0 Discovery of particle having mass less than predicted mass
Recent work at SLAC confirmed discovery of a new particle dubbed as Ds (2317) having mass 2,317 mega-electron volts. But this mass is far less than current estimates, is a mathematical puzzle [7]. This discrepancy can be explained with help of equation D E = Ac² D M with value of A more then one.
The annihilation of matter and antimatter or vice-versa is explained by D E =D mc² and experiments are being continuously conducted in this regard [12]. In case at some stage more anomalies (i.e. magnitude of mass converted into energy in annihilation of matter and antimatter or vice-versa) are observed less or more than predicted by D E =D mc²are observed then it would further serve as an evidence in favour of D E = Ac² D M. Thus this equation acts as scientific stimulant. In brief the comparison and conclusions of equations D E = Ac² D M and D E =D mc² are given on Table III.

Table III

Sr. No	Value of A in D E = Ac² D M	Comparison	Applications.
1	A=1	D E = Ac² D M = D mc²	Nuclear reactions
2	A>1	D E >D mc²	Gamma Ray Bursts, Quasars, Supernova etc.
3	A<1	D E < D mc²	Earliest origin of the universe, M =10⁵⁵kg , D E =10^-4444 J or less. Possibly in chemical reactions
4	A=0	D E = D m =0 or tends to 0	Neutron hits ₂₆Fe⁵⁶, no reaction takes place.

Acknowledgements
Author is highly indebted to Dr. Belinda Wilkes for sending useful scientific information about black holes and quasars. Thanks are due to Prof. Ramaswamy for sending original research papers of Einstein and numerous other critics for constructive comments.

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