ENERGY & MOMENTUM

 

 

 

James A. Putnam

 

 

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The empirical measurements of changes of velocity are the starting point from which theoretical physics is derived. Fundamental theory consists of mechanical interpretations of causes for changes of velocity. In general this mechanical interpretation is called ‘force’. There are some theoretical interpretations that avoid the use of the word force. However, in this essay, anything that causes change of velocity is understood to be generic in its empirical support. In all cases of an object changing its velocity some unknown cause is involved. So long as the natures of causes are unknown and their effects continue to be changes of velocity, it remains empirically proper to group them together under the general category of ‘force’.

 

When an identical force is applied to a variety of objects, it is observed that the objects change their velocities by a variety of magnitudes. In other words, each object has its own magnitude of resistance to force. This resistance to force is called ‘mass’. All higher-level theory consists of mechanical interpretations of force, resistance to force and their relationships to distance and time.

 

Newton’s formula expressing the general relationship between force, resistance to force and acceleration is:

 

 

This equation says force is equal to mass multiplied by acceleration.  Written in a slightly different form:

 

 

It is seen that acceleration, or change of velocity measured with respect to time, is equal to force divided by resistance to force. The left side of this equation represents empirical evidence. The right side represents properties that are recognized to exist, but have unknown physical natures.

 

Newton’s formula should be read as: Something, of an unknown nature, causes an object to accelerate an amount inversely proportional to a resistance of unknown nature. It is not proper to say: A force causes a mass to accelerate. The object is not the mass. The object is not in the equation. Only its properties enter into mathematical analysis. The equation addresses an effect and unknown causes.

 

It is important to recognize from the start that the only physically observed quality represented in the equation is acceleration. Since force and mass are unobservable, they cannot be used for the derivation of a known physical nature for anything. This lack of knowledge is actually the basis for the need to develop theory. Theory begins with mechanical interpretations of the causes for acceleration and resistance to acceleration. The problem of lack of knowledge enters at the beginning of theory and remains pervasive through all of physics theory.

 

The combining of force and resistance to force with distance and time is the basis for the derivation of the rest of physics theory. This practice begins with assigning unique units to both force and mass. These units are theoretical because there is no empirical knowledge to tell us how to proceed. We do not begin by knowing, so we begin by assuming. It is assumed that force and mass require unique units of measurement. This practice is acceptable and useful so long as it is recognized as being temporarily necessary and to be improved upon as soon as possible.

 

Higher-level, theoretical, mechanical properties consist of equations formed from different combinations of force, mass, distance and time. Beginning with force multiplied by time. This calculation is called momentum as represented by P:

 

 

 

Newton’s  f = ma follows from the use of this property in his equation:

 

 

This says force is equal to the rate of change of momentum with respect to time. Multiplying both sides by the differential measure of time dt:

 

 

The differential of time means an extremely small period of time. Differentials, in general, represent almost instantaneous changes of a varying function. They are the basis of the derivation of the mathematics of change called calculus. Force multiplied by dt is the differential of momentum as is symbolized by the right side of the equation. Integrating the differential of momentum, with initial conditions of zero, in other words v = 0 when t = 0, yields:

 

 

This mathematical property is conserved in nature. It is used in the solutions of many problems. It is convenient to give it a name, so It is called momentum. The assignment of this name adds no new physics knowledge. The quality remains defined as force times time or ft and, for non-relativistic events, is found to be equal to the constant mass of an object times its final velocity.

 

 

The integral of force times the differential of time is the calculus definition of a change in momentum. It does not matter how the force may have been varying with time. The end result is equal to mass times the final velocity. Since this is true for all variations of force times time, then any calculation of momentum can be represented by a constant force times the time period that yields the same result. This is written as:

 

 

It says a constant force applied over a period of time causes an object with mass m to achieve a velocity of v.

 

Momentum is just one method of measuring the effect of a change of velocity. There is a second method of measuring the same effect. It is to measure the force as being applied across a distance. This calculation is represented by:

 

   

 

This calculation of force times the differential of distance is the differential of kinetic energy. Energy is also conserved and is very useful in solving many problems. It is often used in conjunction with momentum to help solve problems of motion. There are no other kinds of energy or momentum. All energy and momentum properties are simply calculations describing an object that has changed its velocity. Both energy and momentum involve action. There is no state of inaction that fits with their definitions.

 

Potential energy is not energy. It is a recognition that a force exists. If that force is freed to act upon an object across a distance, then there will exist the action upon which the calculation we call energy can be performed. In other words, there is the potential for an object to change its velocity and for a new product of force times distance to be calculated. There simultaneously exists, just as valid, the potential for momentum to be calculated.

 

Energy is not superior, by virtue of a unique material existence, to momentum. They are two perspectives of an identical event. The magnitudes of the calculations of energy and momentum for the same event will differ. However, since they are measuring the same event, they should be expected to share similarities such as conservation.

 

Energy and momentum are related to each other by the factor:

 

 

Or, saying the same thing:

 

 

It can be seen that the distance x can be calculated using:

 

 

And:

 

 

Therefore, momentum is converted to energy by multiplying the momentum by the distance traveled divided by the period of time. The distance divided by the time and then multiplied again by the time simply yields the distance back.

 

 

The distance times the force is the energy.

 

So, both energy and momentum are calculations involving the same force whose effect is measured with regard to either distance or time. In other words the nature of either energy or momentum is dependent upon the nature of force. It is force that is the property that must yet become understood. Once it is known, then its calculations with either distance or time become mere mathematical quantities.

 

However, in modern physics theory both momentum and energy are interpreted as if they are unique fundamental physical properties that have their own physical substance. Energy is given a position of prominence by interpreting it as the physical substance from which all things are formed. The universe never gains or loses energy. Its supply of energy is transferred around and converted into different forms. This idea cannot be derived from theoretical fundamentals. It is introduced as a ‘given’ in higher-level theory.

 

Empiricism is the basis of the fundamentals. If a material nature for energy were shown to exist by empirical evidence, then it would enter physics theory at the beginning fundamental level. As far as the empirical universe is concerned, the name of energy could be replaced with the phrase force times distance and nothing of empirical significance would be lost.

 

When it is said that mass and energy are equivalent, then it is empirically more informative to restate this as: Mass and the product of force times distance are equivalent. However, this statement is still not expressed in full empirical form. To state the same claim fully in empirical language it is necessary to say: Resistance to force and force times distance are equivalent. However, whether or not this statement is true depends upon understanding the significance of velocity in the equation. If velocity simply represents a value of conversion, then the statement may represent the truth.

 

If velocity plays its own important physical role in the operation of the universe, then its own significance must be recognized and the statement of simple equality needs to be reconsidered and examined for deeper more complete meaning. The importance of this point becomes clear when including the speed of light. The speed of light is a very important property of the operation of the universe. Using it as a mere value of conversion may be undercutting its true significance when considering a relationship between mass and force times distance.

 

The modern theoretical interpretation of energy existing as a fundamental physical material is represented by Einstein’s equation:

 

 

This equation describes total energy. It was derived, by Einstein, as a part of his equation for kinetic energy. His complete equation is written as:

 

 

This says kinetic energy is equal to total energy minus rest energy.

 

This is the equation used to support the statement that mass and energy are equivalent. It is used to predict the conversion of energy into matter. However, mass and matter are not the same. Mass is a property of matter. Matter is the theoretical conclusion that a material substrate exists from which properties of force and resistance to force emanate. The existence of matter cannot be empirically demonstrated. However, we know, from patterns in changes of velocity that objects, whatever their true nature, do have properties. The properties are empirical. Whether or not there exists a material basis in addition to the properties themselves does not effect the derivation of physics theory.

 

If energy and mass are really equivalent, then wherever there is energy there also is mass. Light would have mass. Yet it is often stated that light does not have mass. The theoretical basis of this claim is represented mathematically by converting Einstein’s energy equation into an equation of energy and momentum. This conversion produces an equation that can appear to not directly include mass. The equation is written as:

 

 

This equation permits the theorist to speak in terms of energy and momentum instead of mass and velocity. With regard to light there is no rest energy so the equation becomes:

 

 

Now it can be said that light has both energy and momentum. Why does this eliminate a dependence upon mass? It only does so for those theorists who believe in the independent existences of energy and momentum. By this practice, the derivation of theory is turned upside down. Instead of working from empirical evidence and developing theory based firmly upon fundamentals, the theorist assumes the unique material existence of energy and momentum are established facts. Now the theorist can begin with the givens of energy and momentum and work backward to derive such things as mass and force.

 

This practice releases higher-level theory from being dependent upon the fundamentals. Instead the fundamentals become dependent upon higher-level theory. An example of the consequences of this change in direction of theory is: [Every single principle that we teach in intro college physics is based on only two principles: Conservation of momentum and conservation of energy/mass. … Conservation of momentum is based on the isotropic symmetry of empty space, conservation of energy on the symmetry of time….] In other words, the fundamentals of physics are to be based upon properties that are empirically impossible to verify. These properties are the isotropic symmetry of empty space and the symmetry of time.

 

Energy becomes something greater than force time distance. It is assumed to be a universal fundamental material that is moved around and transformed. Its greatest theoretical achievement is it can produce force and mass. So force and resistance to force are no longer the beginning point for physics theory. They become the result of a reverse derivation beginning with a ‘fundamental given’ called energy. Unfortunately for theoretical physicists, no one can produce one thimble full of energy. In spite of this empirical deficiency, for modern theory, energy is considered a key to establishing the correctness of physics theory.

 

The cornerstone of this belief is provided by Einstein’s energy equation:

 

 

This equation says that kinetic energy is equal to total energy minus rest energy. Since energy and momentum are considered to be important properties by which to understand the nature of the universe, it has been the practice to express energy in terms of momentum. In the fundamentals of relativity theory, this can be accomplished in the following manner. Mass, according to Einstein, is expressed as:

 

 

Squaring this equation:

 

 

Multiplying both sides by:

 

 

Yields:

 

 

Multiplying yields:

 

 

Rearranging gives:

 

 

This equation can be symbolically represented using general symbols for total energy, rest energy, and momentum:

 

 

It says that total energy squared equals the term momentum squared times the speed of light squared plus the term rest energy squared.

 

This equation is derivable from the fundamentals so long as it is applied to the motion of particles moving at less than the speed of light. These are the particles we actually observe and from which we gather our empirical evidence. The P in the formula is derived only as the product mv. It is dependent upon mass. Photon energy is where a break with the fundamentals is made. When the equation is applied to photons, it is done in such a way that its fundamental derivation no longer applies to its use. First, since photons are never at rest, they are said to have no rest energy. In this case the equation reduces to:

 

 

Our ignorance about the nature of mass is what leads to this confusion over how to apply Einstein’s energy equation to photons. Einstein’s equation for mass does not help us. When the particle speed is C, there is no solution for the quantity of mass. The equation becomes indeterminate. What then is the most fundamental, empirically based form we can use to express what this equation is telling us? It is to replace the terms for both energy and momentum with their fundamental definitions. That is to say:

 

 

The equation contains force times time in place of momentum and force times distance in place of energy. The term f₀ represents the intrinsic force of an object at rest that we know is available as in nuclear reactions.

 

Using this equation, we need no longer to be bogged down in the debates over what is mass and what has mass. Instead we can talk in terms of time, distance and force. For a photon the available rest force is zero. The energy equation for a photon reduces to:

 

 

This equation does represent empirical evidence. We observe that photons do exert force over a period of time. That is how they cause particles to change their velocities.

 

What then may be the significance of the speed of light in the equation:

 

 

It appears to represent the constant of proportionality between energy and momentum. In other words, it represents the conversion of force times distance to force times time. This conversion is accomplished by multiplying force times distance by the velocity of the photon. The velocity C is necessary instead of the conventionally expected (1/2)C because the velocity of the photon is always C. Therefore the distance across which the photon acts is always equal to the velocity C multiplied by the period of time required for the photon to cause its effect upon a particle of matter. The above equation should be written as:

 

 

This equation is representative of a property of photons. It applies to an object moving at the speed of light. It is a definition of potential energy. It is not energy because it is not representative of force acting across a distance. It is representative of the potential for force to act across a distance. There is also potential momentum. The product of force times time is representative of the potential for the force to be applied during a period of time. Later when the photon has applied its force causing an object to change its velocity, there is the opportunity for both measurements to be made. The result of the measurements can be used to calculate energy and momentum. 

 

Thus far, I have used the speed of light in the manner usually employed. It is usually said to be the conversion multiplier between mass and energy. For example, It greatly magnifies the amount of energy said to be intrinsic to rest mass:

 

 

 

However, there is something about this interpretation that suggests possible severe inadequacy. The speed of light is an incredibly important fundamental property of the universe. The inclusion of the square of the speed of light in Einstein’s energy equation must be the result of some very important fundamental property. 

 

It is telling us that matter and the properties of light are interdependent. Something about matter behaves in a manner similar to that of photons. It can be anticipated that we are to discover that matter is representative of the potential for energy to be transmitted at the speed of light. Matter produces light and does this at the speed of light. There must be even more to it.

 

The light leaves matter moving away at a controlled speed. It must then be the case that matter is directly related to the cause for the existence and speed of light. Matter cannot be wholly unique from the nature of light. It is representative of potential light. Its own nature shares unity with the nature of light. Matter makes light active, and light makes matter active. They share properties and abide by shared principles. This interdependence of properties demonstrates fundamental order. Order is caused by fundamental unity.

 

Whether or not this speculation proves to have merit, we are forced to admit that the establishment of both energy and momentum as having unique physical presence is speculative. The true relevance of the properties called energy and momentum are dependent upon gaining an understanding of the property of force. So long as we do not understand the nature of force, we cannot hope to understand mass, momentum or energy. Learning the natures of force and resistance to force is essential. In my theory, I pursue new interpretations for the natures of force and resistance to force.