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Research Communication
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The Negative Gravitational Mass
Lee Jiang
Sydney Branch,
Science Platform Coorporation, Austalia
Abstract - The concept of negative mass was dropped a long time ago because it appeared to
conflict with Newton’s Second Law of Motion and also no material object has ever been found to have
a negative mass which can be shown by experiment. However, after studied Newton’s Second Law, it
would not contradict the concept of negative mass if applied in the proper way. The misconception
came from the term ‘Negative Mass’. In physics, it should be termed the “Negative Gravitational
Mass”.
Keywords: Newton’s Second Law of Motion, Negative Mass, Negative Gravitational
Mass .
1. Introduction
The concept of a negative mass has been implied from mathematical models such
as the formulae of physics equations.
In employing mathematical models to describe nature, we come across solutions to
equations that result in negative masses. For example, the formula for the energy of a
relativistic particle is
E2 = m2c4 + p2c2
(1)
So a particle with a certain positive energy but no momentum could presumably
have a positive or negative mass. Dirac interpreted these negative mass states as antiparticles that he hid away in the 'Dirac Sea.' While this outdated interpretation still
lives on in old-textbooks and new-age books on quantum mechanics, we now know
that this picture is wrong. Anti-particles have positive masses just as any other particle
[6]. These theoretical 'negative' solutions are simply not found in practice and the
concept of them has been dropped. Not everything that has mathematical meaning has
physical meaning! [1]
Another place where people like to talk about negative masses is in reference to
'tachyons.' The tachyon (whose name comes from the Greek word tachys for swift)
was originally any solution to Special Relativity that produced a velocity greater than
the speed of light. Such a state (like Dirac's 'negative energy' states) is non-physical.
These particles would have imaginary masses (that is the mass-squared is negative)
and this is just as unphysical as a negative mass or a negative energy. Thus such
solutions are always discarded or removed from any theory that claims to describe
Nature. [1]
2. Negative Gravity
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Although no known particles have negative mass, physicists (primarily Robert L.
Forward) have been able to describe some of the anticipated properties such particles
would have. Negative mass would produce "negative gravity" that repels ordinary
positive mass, but would be attracted to other negative mass particles in a normal
matter.
F = G(m1m2)/r2
(2)
For a negative value of m1 with positive value of m2, F is negative (repulsive). At
first glance it would appear that a negative mass would accelerate away from a
positive mass, but by Newton's law:
F = ma
(3)
If m is negative and F is negative, then a must be positive - the negative mass
accelerates toward the positive mass. If m is positive, however, and F is still negative,
then a is negative and the positive mass accelerates away from the negative mass. It
can be shown that if m1 and m2 are of equal but opposite mass, then the combined
system of positive and negative particles will accelerate indefinitely without any
additional input into the system.
This behavior is bizarre, in that it's completely inconsistent with our 'normal
universe' in the context of the, normally encountered, behavior of positive masses, but
it is completely mathematically consistent and introduces no apparent contradictions
when physics analysis is performed on the behaviors. [2,5]
In physics, negative mass is mass with negative properties. One of the properties is
that negative mass is mass that repulses all other mass. The negative matter is a
mathematical construction, and, most likely, not a physical reality. [3]
3. Physics of Mass
As the above reasoning used mass, Newton’s Second Law and the Law of
Universal Gravitation, it is necessary that we review the physics definitions of them
[4]. Then based on the definitions, we will apply the correct solution to the negative
mass.
3.1. Mass
The most obvious external force acting on an object is its weight. Mathematically:
W = mg
(4)
The mass m in the equation is called the gravitational mass of the object, which is
measured statically. ‘g’ is the gravitational field strength. On the surface of Earth, g is
9.8ms2. The equation shows that “heavy” objects have more weight (W).
3.2. Newton’s Second Law of Motion
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Isaac Newton (1642-1727) expressed what he knew about force in three laws –
these laws are known, worldwide, today as Newton’s Laws of Motion.
A brief statement of the Second Law is: “The acceleration of an object is
proportional to the unbalanced force acting on it and is inversely proportional to its
mass”. This can be expressed as:
F = ma
(5)
The m is called Inertial Mass, which is measured dynamically (that is, while
moving). Extremely accurate experiments have shown that the gravitational mass is
equal to the Inertial Mass.
3.3. Newton’s Law of Universal Gravitation
From Kepler’s law of Periods, Newton deduced that gravitational force obeys an
inverse square law and from Kepler’s law of Areas, he deduced the force to be a
central force (that is, it always acts along the line joining the Sun and the planet).
Combining these ideas Newton proposed his Law of Universal Gravitation: “A
gravitational force of attraction acts between any two objects in the universe. The
force between these objects of mass m1 and m2 whose centres are separated by a
distance d is proportional to the product of their masses and inversely proportional to
the square of their separation”. That is
F = G(m1m2)/d2
(6)
Here F is the force, G is a constant called the “universal gravitational Constant”, m1
and m2 are the gravitational masses of the objects and d is the separation between the
centres of the two objects.
4. The Negative Gravitational Mass
In the above definitions, we found that the concepts of gravitational mass and the
inertial mass are different. The gravitational mass is measured statically, and the
inertial mass is measured dynamically. If we only consider positive mass, then the
masses are found to be identical. However, for the negative mass, it is a different story.
The name negative mass is misleading and causes confusion because it does not
clearly state whether it is a negative gravitational mass or negative inertial mass. From
the definition of Newton’s Second Law, the inertial mass is just a constant value to
balance the equation between force and acceleration. It is always positive. If we
assume that it is the same as gravitational mass, it is only equal to the absolute value
of the gravitation mass. In this case, Newton’s Second Law can be modified to give:
F =| m|a
(5)
Here m can be taken as the gravitational mass.
To be accurate and correctly define the negative mass, it should be called the
“negative gravitational mass”. Then apply it back to negative gravity [2,5], and we
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find Newton’s Second Law supports the concept of negative gravity. There is no
conflict at all.
5. Conclusion
There was a misunderstanding about gravitational mass and inertial mass that led to
the rejection of negative mass. If we clear the concept and define the “negative
gravitational mass” properly, we can make a small modification to Newton’s Second
Law to make it support negative gravitational mass. Therefore, “negative gravitational
mass” is not just a mathematical concept because it can be shown not to be in conflict
with the laws of physics.
6. Acknowledgment
The author would like to thank Dr. Y.J. Wang providing support to the work.
7. References
[1] B. Nelson, “How can something have a negative mass, and what does that mean?”,
www.physlink.com/Education.
[2] The Free Dictionary, “Negative mass”, encyclopedia.thefreedictionary.com.
[3] Memory
Alpha,
the
free
Star
alpha.org/en/index.php/Negative_mass.
Trek
reference,
“Negative
mass”,
www.memory-
[4] N. Warren, “Excel Preliminary Physics”, Pascal Press 2001.
[5] P. V. Bradford, “Alternative ways of looking at Physics”, www.concentric.net/~Pvb/physics.html.
[6] B. Nelson, “If there is anti-matter, couldn't there be anti-energy?”, www.physlink.com/Education.
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