Can mass be negative? Текст научной статьи по специальности «Философия, этика, религиоведение»

Mathematical Structures and Modeling 2018. N. 1(45). PP. 43-48

UDC 531.6 DOI: 10.25513/2222-8772.2018.1.43-48

CAN MASS BE NEGATIVE?

Vladik Kreinovich1

Ph.D. (Phys.-Math.), Professor, e-mail: vladik@utep.edu Sergei Soloviev2 Ph.D. (Phys.-Math.), Professor, e-mail: sergei.soloviev@irit.fr

1 University of Texas at El Paso, El Paso, Texas 79968, USA 2Institut de Recherche en Informatique de Toulouse (IRIT), 31062 Toulouse, France

Abstract. Overcoming the force of gravity is an important part of space travel and a significant obstacle preventing many seemingly reasonable space travel schemes to become practical. Science fiction writers like to imagine materials that may help to make space travel easier. Negative mass - supposedly causing anti-gravity - is one of the popular ideas in this regard. But can mass be negative? In this paper, we show that negative masses are not possible - their existence would enable us to create energy out of nothing, which contradicts to the energy conservation law.

Keywords: negative mass, equivalence principle, anti-gravity, energy conservation law.

1. Formulation of the Problem

Overcoming the force of gravity is an important part of space travel and a significant obstacle preventing many seemingly reasonable space travel schemes to become practical. Science fiction writers like to imagine materials that may help to make space travel easier. Negative mass - supposedly causing anti-gravity - is one of the popular ideas in this regard. But can mass be negative?

2. About Our Approach

The considerations below are based on existing fundamental concepts and well known experiences. For example, we assume:

that the masses add up when we combine two bodies, that forces that act on a solid body add up, that it is always possible to rigidly bind two bodies together, that for every body, one can construct an anti-body with the same mass, etc. So, strictly speaking, we consider the possibility of a negative mass in the context of these concepts and experiences. Our conclusion is that the negative mass is incompatible with these concepts and experiences - and incompatible on a very basic level.

Comment. It is worth mentioning that while we take Newton's formula for the gravitational force as given, this formula itself can be derived from the fundamental assumptions - like additivity of masses and forces; see, e.g., [3].

3. Reminder: There Are Different Types of Masses

To properly answer the question of whether negative masses are possible, it is important to take into account that there are, in principle, three types of masses:

• inertial mass mi that describes how an object reacts to a force F : the object's acceleration a is determined by Newton's law mi • a = F ; and

• active and passive gravitational masses mA and mp: gravitation force exerted by Object 1 with active mass m ai on Object 2 with passive mass mp 2 is equal to F = G • mA1 2, where r is the distance between the two objects; see, e.g., [1,4]. f

Since F = mi • a, the formula for the gravitational force can be rewritten as

mAi • mp2 , , mi2 • a,2 = G--2-. (1)

Can any of these masses be negative?

4. All Three Masses Are Proportional to Each Other

General idea. To answer the above question, let us recall that, due to energy conservation and the properties of anti-particles, all three masses are proportional to each other; see, e.g., [2]. For completeness - and to make sure that the corresponding arguments are applicable to negative masses as well - let us recall the corresponding arguments.

Active and passive masses are proportional to each other: case of positive masses. Let us first show that the active and passive masses are always proportional to each other, i.e., that

mAi _ mA2 mPi mP2

for every two objects. We will first show it for bodies of positive mass. Indeed, suppose that for some pair of bodies, this is not true, i.e.,

mAi _ mA2 mP i mP 2'

Multiplying both sides of this inequality by both passive masses, we conclude that mA1 • mP2 _ mA2 • mP 1. Thus, the gravitational force exerted by Object 1 on Object 2 is different from the gravitational force exerted by Object 2 on Object 1.

So, if we combine these two objects by a rigid rod, the overall force acting on the resulting 2-object system would be different from 0. Thus, if this system was originally immobile, it will start moving with a constant acceleration. We can

then stop this system, use the gained kinetic energy to perform some work, and thus, get back to the original configuration - with some work done. We can repeat this procedure as many times as we want. This way, without spending anything, we can get as much work done as we want (and/or as much energy stored as we want). This possibility to get energy from nothing, without changing anything, clearly contradicts to energy conservation law, according to which such perpetuum mobile is impossible.

This contradiction shows that for positive masses, active and passive masses should be proportional to each other: mA = c • mp for some constant c. If we plug in this expression for the active mass into the formula (1) for gravitational force, we conclude that

mP i • mp2 mj2 • a.2 = G • c •-—-,

i.e.,

mP i • mP2

mi2 • a,2 = G' •

where we denoted G' = G • c. Thus, we get the formula similar to the formula (1), but with the passive and active masses equal to each other.

So, we can conclude that when masses are positive, active and passive masses are equal: mA = mp.

Active and passive masses are proportional to each other: general case.

What if at least one of the masses - either active or passive - is negative? In this case, the argument about gaining energy does not necessarily apply: e.g., when the product mp • mA is negative, the 2-object system does not gain energy, only loses it.

A slight modification of this thought experiment, however, enables us to gain energy. Indeed, let us consider an object with different active and passive masses mA = mp. Instead of considering this object on its own as before, let us attach it to another object with a big positive mass Ma = MP > max(|m^|, |mp|). This combination has active mass Ca = mA + Ma and passive mass Cp = mp + MP. Since MA = MP > max(|m^|, |mp|), both these combined masses are positive. Since MA = MP and mA = mP, we conclude that Ca = CP. So, the active and passive masses of the combined object are positive and different - and we already know that this leads to a contradiction with the energy conservation law. So, for negative masses, active and passive gravitational masses are also always equal.

Since the active gravitational mass is always equal to the passive gravitational mass, in the following text, we will simply talk about gravitational mass mG.

Gravitational and inertial masses are proportional to each other: case of positive masses. The important property that we will use is that any type of matter, when combined with the corresponding antimatter, can annihilate, i.e., get transformed into photons, and these photons can get transformed into some other types of matter. For example, we can start with iron and anti-iron, annihilate them, and then get gold and anti-gold. We will also take into account that experiments seems to confirm that matter and corresponding anti-matter have the same

inertial and gravitational properties, in particular, the same value of the inertial and gravitational mass; see, e.g., [4].

We want to prove that for all materials, the ratio ^^ of gravitational and inertial

mi

masses is the same. Indeed, let us assume that there exist two materials for which

this ratio is different, i.e., for which _ ^^. Without losing generality, we

mn mi2

can assume that the ratio is smaller for the first material: - < -. This

mn mi 2

means that if we select two objects of the same inertial mass mi 1 _ mi2 from the first material and from the second material, then the gravitational mass of the first object is smaller: mG1 < mG2.

We can then get the following scheme for getting energy out of nothing. We place a body and an identical anti-body of the first material at some distance from the gravitational attractor of some mass M - e.g., from the Earth. We then move both bodies a small distance h away from the Earth. The corresponding force is

F _ G • '2mG12-, thus the energy that we need to spend for this move is equal to

2mG1 • M

F h _ G h

Once we reached the distance r + h, we annihilate both objects, and use the resulting photons to create a pair of a body and anti-body of material 2. Then, we move the new object back to the distance . This way, the force is equal to

F _ G • ——2-, thus the energy that we gain is equal to

2mG2 • M

F h _ G h

At the distance r, we annihilate both objects, and use the resulting photons to create the original pair of the body and anti-body of Material 1.

Now, we are back to the original state, but, since mG2 > mG1, we gained more energy that we spent - i.e., as a result, we get energy out of nothing. The impossibility of such a perpetuum mobile shows that, at least for positive masses, gravitational and inertial masses should be proportional to each other: mG _ const •mi. Thus, if we select the same unit for measuring both gravitational and inertial masses, we can conclude that when masses are positive, gravitational and inertial masses are equal mG _ mi.

Gravitational and inertial masses are proportional to each other: general case. What if at least one of the masses - either gravitational or inertial - is negative? In this case, the above argument does not necessarily apply: e.g., if the inertial mass of some material is negative, we cannot transform it into a material with a positive inertial mass.

A slight modification of this thought experiment, however, enables us to gain energy. Indeed, let us consider an object with different gravitational and inertial masses mG _ mi. Instead of considering this object on its own as before, let us

attach it to another object with a big positive mass MG = Mi > max(|mG|, |mj1). This combination has gravitational mass CG = mG + MG and inertial mass Ci = mi + Mi. Since MG = Mi > max(|mc|, |toj1), both these combined masses are positive. Since MG = Mi and mG = mi, we conclude that CG = Ci. So, the combined object has positive and different gravitational and inertial masses - and we already know that this leads to a contradiction with the energy conservation law. Since the gravitational mass is equal to the inertial mass, in the following text, we will simply talk about the mass m.

Comment. The fact that all the masses are proportional to each other is one of the formulations of the famous equivalence principle - one of the main underlying principles of Einstein's General Relativity; see, e.g., [1,4]. Note that in our analysis, we did not use this principle - we derived the proportionality of different types of masses from fundamental concepts.

Conclusion: unfortunately, there is no such thing as anti-gravity. An unfortunate conclusion is that for every object, whether its mass m is negative or positive, its acceleration in the gravitational field of a body of mass M is determined by the

formula m • a = G • —, thus a = G • . This acceleration does not depend on

the mass of the attracted body - so all objects follows the same trajectory, negative masses same as positive ones.

5. So Are Negative Masses Possible?

Finally, we can answer the question of whether negative masses are possible. Suppose that negative masses are possible. Then, by attaching an object with a negative mass m < 0 to a regular object with a similar positive mass |m| = —m, we get a combined object whose overall mass M is 0 (or at least is close to 0). Since the mass M is close to 0, even a very small force F will lead to a

F

huge acceleration a = —. Thus, without spending practically any energy, we can

accelerate the combined object to as high a velocity as we want. Once the object reaches this velocity, we dis-attach the negative-mass object - let it fly away. As a result, we now have an object of positive mass |m| with a very large kinetic energy - and we can use this energy to perform useful work.

This scheme is not as clear-cut as the previous schemes, since here, we do not exactly go back to the original state - we lose a negative-mass body. However, we can do this for negative-mass body of arbitrarily small size - and still gain a lot of energy. Thus, while we cannot gain energy and get back to exactly the same original state, we can get back to a state which is as close to the original state as we want - and still gain as much energy as we want. This clearly contradicts to the idea of energy conservation. Thus, negative masses are not possible.

Acknowledgments

This work was supported in part by the National Science Foundation grant HRD-1242122 (Cyber-ShARE Center of Excellence).

References

1. Feynman R., Leighton R., Sands M. The Feynman Lectures on Physics. Boston :Addison Wesley, 2005.

2. Kreinovich V. Astronomical Tests of Relativity: Beyond Parameterized Post-Newtonian Formalism (PPN), to Testing Fundamental Principles // Relativity in Fundamental Astronomy / S. Klioner, P.K. Seidelmann, M.H. Soffel (eds.). Proceedings of IAU Symposium. No. 261. Cambridge : Cambridge University Press, 2009. P. 56-61.

3. Kreinovich V., Sriboonchitta S. Quantitative justification for the gravity model in economics // Predictive Econometrics and Big Data / V. Kreinovich, S. Sriboonchitta, N. Chakpitak (eds.). Cham : Springer Verlag, 2018. P. 214-221.

4. Misner C.W., Thorne K.S., Wheeler J.A. Gravitation. San Francisco : Freeman Publ., 1973.

МОЖЕТ ЛИ МАССА БЫТЬ ОТРИЦАТЕЛЬНОЙ?

В. Крейнович1

к.ф.-м.н., профессор, e-mail: vladik@utep.edu С. Соловьев2

к.ф.-м.н., профессор, e-mail: sergei.soloviev@irit.fr

1 Техасский университет в Эль Пасо, США 2ИТ-исследовательский институт Тулузы (IRIT), 31062 Тулуза, Франция

Аннотация. Преодоление силы тяжести является важной частью космических путешествий и значительным препятствием, мешающим многим, казалось бы, разумным схемам космических путешествий стать практичными. Писатели-фантасты любят придумывать материалы, которые могут облегчить космические путешествия. Отрицательная масса — предположительно вызывающая антигравитацию — является одной из популярных идей в этом отношении. Но может ли масса быть отрицательной? В этой работе мы показываем, что отрицательные массы невозможны — их существование позволило бы нам создать энергию из ничего, что противоречит закону сохранения энергии.

Ключевые слова: отрицательная масса, принцип эквивалентности, антигравитация, закон сохранения энергии.

Дата поступления в редакцию: 27.11.2017

Mathematical Structures and Modeling 2018. N. 1(45). PP. 49-51

UDC 531:372.853 DOI: 10.25513/2222-8772.2018.1.49-51

THE SUMS OF Mi • Vi AND Mi • Vf ARE PRESERVED, WHY NOT SUM OF MI • V?: A PEDAGOGICAL REMARK

John McClure

Ph.D. (Eng.), Professor, e-mail: mcclure@utep.edu Olga Kosheleva Ph.D. (Phys.-Math.), Associate Professor, e-mail: olgak@utep.edu

Vladik Kreinovich Ph.D. (Phys.-Math.), Professor, e-mail: vladik@utep.edu

University of Texas at El Paso, El Paso, Texas 79968, USA

Abstract. Students studying physics sometimes ask a natural question: the momentum - sum of mi ■ Vi - is preserved, the energy - one half of the sum of mi ■ vf - is preserved, why not sum of mi ■ v?? In this paper, we give a simple answer to this question.

Keywords: momentum conservation law, energy conservation law, teaching physics.

1. Formulation of the problem

Students studying physics sometimes ask a natural question:

• the momentum - sum of mi ■ Vi - is preserved,

• the energy - one half of the sum of mi ■ vf - is preserved, so

• why not sum of mi ■ v??

In this paper, we give a simple pedagogical answer to this question.

2. Our explanation

To answer the above question, let us consider a simple 2-body 1-D mechanical problem. We have two small (point-like) solid objects with masses mi = 2 and mi = 1 on a straight line, the second object is to the left of the first one. Originally:

• the first object is at rest (its velocity is vi = 0), while

• the second object moves towards the first object with velocity v2 = 1. What will happen when the second object hits the first one?

The situation is invariant with respect to rotations around the line connecting the two objects, so the resulting trajectories will also be invariant - and thus, both objects will continue to move along the same line. The only remaining question is: what will be the new velocities v'i and v2?

We know that the momentum is preserved, and we know that kinetic energy is preserved. Thus, we can conclude that

mi ■ v! + m2 ■ v'2 = mi ■ vi + m2 ■ v2

(1)

and

mi • (vi)2 + m2 • (^2)2 = mi • v\ + m2 • (2)

Substituting the known values mi = 2, m2 = 1, vi = 0, and v2 = 1 into these formulas, we conclude that

2v[ + v2 = 1 (1a)

and

From (1a), we conclude that

2K )2 + (v2 )2 = 1. (2a)

4 = 1 — 2v[. (3)

Substituting this expression into the formula (2a), we get

2(4 )2 + (1 — 2v[ )2 = 1.

Opening the parentheses, we get

2(v[ )2 + 1 — 4v[ + 4(^ )2 = 1.

Subtracting 1 from both sides and bringing similar terms together, we get

6(v[ )2 — 4v[ = 0,

v[ • (6v[ — 4) = 0.

i.e., equivalently,

Thus, we have two options:

• either v'1 = 0,

12

• or 6v' — 4 = 0, so that v\ = -.

i . i 3

In the first case, when v\ = 0, from the formula (3), we conclude that v'2 = 1. This means that the the first object remains immobile, and the second object continue with the same speed - so that it somehow passed through the first object. For solid objects (not for ghosts), this is not possible.

Since the case v[ = 0 is not physically possible, we are left with the second 21

option v[ = -. In this case, the formula (3) implies that 13

41

v'2 = 1 — 2v\ = 1 — = —. 2 1 3 3

One can check that in this case, the equalities (1) and (2) - describing preservation of the sums of mi • Vi and mi • v\ - are satisfied, while the sum of mi • if are not preserved: indeed, in this case,

mx • v\ + m2 • vl = 2 • 03 + 1 • 13 = 1,

while

mi • W)3 + «2 • (4)3 = 2 •(3) 3 + (—3)