## Conservation Laws

### BACKGROUND

In this section, the consequences of unbalanced force generation are examined with respect to the current understanding of the conservation laws of momentum, angular momentum, and energy. The consequences (changes) to the conservation laws are dependent upon the existence of an unbalanced force. Currently accepted physics states that creation of an unbalanced force is impossible. The Cannae Drive is a device that generates an unbalanced force. The consequences to the conservation laws are required by the existence of the Cannae Drive, or any other mechanism, which is shown to generate an unbalanced force. The consequences to the conservation laws discussed in this section are dependent upon the existence of a demonstrated unbalanced force.

#### CURRENT CONSERVATION LAWS

The three conservation laws that are affected by the generation of an unbalanced force are conservation of momentum, conservation of angular momentum, and conservation of energy. These three conservation laws are tied directly to the concepts of space and time symmetry. A verified unbalanced force necessitates a new understanding of these conservation laws. The conservation laws are listed below.

CONSERVATION OF LINEAR MOMENTUM Conservation of linear momentum is a fundamental law of nature. It states that if no external force acts on a closed system of objects, the momentum of the closed system remains constant.

CONSERVATION OF ANGULAR MOMENTUM states that in a closed system, angular momentum is constant.

CONSERVATION OF ENERGY states that the total amount of energy in a closed system remains constant over time.

NOTE More details on conservation of momentum, conservation of angular momentum, and conservation of energy are available under the "mechanics" tab at Hyperphysics.com.

#### IMPLICATIONS OF A VERIFIED UNBALANCED FORCE

Newton's third law of motion states that for every action, there is an equal and opposite reaction. This law of motion precludes the existence of unbalanced forces. The third law is also fundamentally tied to space and time symmetry. You can read about Newton's third law here. An unbalanced force inherently violates Newton's third law of motion. The existence of an unbalanced force means that there are two classes of force interactions: those that are balanced and those that are unbalanced. The Cannae Drive generates an unbalanced force.

SYMMETRIC-FORCE REACTIONS are reactions where all forces balance and Newton's third law is obeyed.

ASYMMETRIC-FORCE REACTIONS are reactions where at least one unbalanced force exists and Newton's third law is not obeyed. All identified force interactions to date are symmetric reactions. Asymmetric reactions are not known to exist within currently accepted physics. The Cannae Drive generates an asymmetric force reaction by creating an asymmetric reaction between a resonating EM wave and the resonating cavity that contains the EM wave.

#### CONSERVATION OF MOMENTUM

Within the Cannae Drive, a resonating EM field acts asymmetrically on the cavity of the Cannae Drive. The resonating EM field has no net momentum due to interference patterns of the EM wave reflecting over itself. Because the EM wave creates an asymmetric force on the cavity, the cavity changes in net momentum. The EM wave cannot balance the momentum change of the cavity since the resonating EM wave has a zero momentum before, during, and after imparting kinetic energy and momentum to the Cannae Drive cavity. For a propagating EM wave, wave momentum is directly related to wave energy. In a resonating EM wave, changes in wave energy do not affect the wave momentum since the wave momentum is always zero.  The Cannae Drive cavity creates a momentum sink.  The Maxwell's Stress Tensor Appendix of this website contains additional information on the momentum sink within a Cannae Drive.

This website will also post a discussion of Noether's theorem and the changes to the conditions under which the theorem applies (currently, the theorem is applied to all force interactions). The discussion will focus on the potential energy term of the Lagrangian. The discussion will show that a resonating EM wave acting asymmetrically on a body is outside the domain of Noether's theorem.

#### CONSERVATION OF ANGULAR MOMENTUM

Any interaction that creates a linear-force imbalance will also allow a closed system to change its angular momentum. If the vector of the unbalanced force does not pass through the center of mass of the body being acted upon, then the net-angular momentum of the system will change. In addition, the Cannae Drive technology can directly create torques on a Cannae Drive cavity by the design of the asymmetric features of the Cannae Drive cavity.

#### CONSERVATION OF ENERGY

SINGLE-PULSE THRUST EQUATION FOR THE CANNAE DRIVE

Energy is conserved within a single EM-wave cycle of the Cannae Drive. For a single EM-wave cycle, the kinetic energy increase of the cavity (and attached mass) is balanced by a decrease in the resonating EM field energy. Some photon energy of the resonating EM wave is converted into kinetic energy of the cavity.

The single-wave-cycle, kinetic-energy change of a vehicle propelled by a Cannae Drive is given by:

For a single-thrust cycle, the Cannae Drive is accelerated from an initial inertial reference frame v₀ to a new inertial reference frame v1 by a velocity change of ∆v. The velocity change is calculated as:

The kinetic energy imparted to the Cannae Drive during one cycle is:

Therefore, the kinetic energy increase of the vehicle after one wave cycle, and the EM energy depleted from the resonating EM wave (from thrusting alone, additional energy will be depleted from the wave due to ohmic heating of the cavity walls) for each cycle is:

Equation (C3) and single-wave-cycle thrusts of the Cannae Drive obey conservation of energy.

MULTIPLE-PULSE ENERGY Over multiple wave cycles, the energy input from the resonating EM wave of the Cannae Drive is given by the total number of EM wave cycles multiplied by the energy loss from the EM wave per single cycle (assumes steady state conditions during operation):

Equation (C5) is not the kinetic energy increase of the Cannae Drive with reference to the initial inertial frame, v₀ of the device. Kinetic energy of the Cannae Drive with respect to an initial inertial reference frame is as follows:

Over multiple wave cycles, there is a difference between the amount of energy depleted from the resonating EM wave and the total kinetic energy increase of the thrusted vehicle with respect to the v₀ reference frame. The energy difference is equal to Equation (C7) minus Equation (C3).

For a single cycle, N=1, and ∆Eₖ = 0 (energy is conserved). For multiple thrusting cycles N>1 and ∆Eₖ ≠ 0 (energy is not conserved). Over multiple wave cycles the kinetic energy imparted to a Cannae Drive is not equal to the EM-field energy used to create the velocity change of the Cannae Drive.

Equation (C8) is valid for operation of a Cannae Drive with uniform pulses and collinear unidirectional-force vectors. Energy changes created by multi-directional operation of the Cannae Drive and/or periods of varying acceleration/deceleration using the Cannae Drive are calculated by summing individual thrusting segments that are uniformly pulsed and have collinear unidirectional-force vectors.

NOTE The energy consequence of Equation (C8) is completely dependent upon the existence of an unbalanced force. The Cannae Drive, or any other device capable of generating an unbalanced force, will create the energy imbalance described by Equation (C8). During symmetric interactions (balanced force interactions), reaction mass kinetic energy decreases with respect to the v0 reference frame.  The kinetic energy decrease of the reaction mass balances the energy of the closed system for symmetric force interactions. Asymmetric (unbalanced force) reactions do not require reaction mass, and this gives rise to the energy imbalance described by Equation (C8).

CONCLUSION The Cannae Drive creates an unbalanced force and, over multiple resonating EM wave cycles, creates a net change of system energy with respect to a v0 reference frame. The energy imbalance is created by the differential between field energy depletion of the EM wave and kinetic energy changes of the Cannae Drive over multiple EM-wave cylces. An isolated body propelled by the Cannae Drive can change the net energy of the isolated system with respect to reference frames outside the isolated system’s reference frame. Any device capable of creating asymmetric force interactions will also generate this energy imbalance.

#### POWER REQUIREMENTS FOR THE QDRIVE

Designs of the Cannae Drive generate linear unbalanced forces. During a single pulse of a Cannae Drive, energy is conserved. The energy of the resonating EM wave is depleted as the cavity is accelerated. Energy must be added to the resonating EM wave to allow for continuous operation (additional pulse cycles) of the Cannae Drive.

During each successive EM-wave cycle of the Cannae Drive, energy is conserved within the inertial frame of the Cannae Drive. Each pulse of the resonating EM wave performs work on the cavity by accelerating the cavity. The energy depleted from the EM wave for thrusting the cavity during each pulse is given by Equation (C3). Additional wave energy is depleted from the resonating EM wave by ohmic heating in the walls of the cavity. Energy is added to the resonating wave to make up for ohmic losses and to make up for the EM field depletion due to imparting kinetic energy to the Cannae Drive. In order to maintain constant acceleration for a constant mass, the power required to accelerate mass during a single wave pulse increases linearly with pulse length according to Equation (C9) below:

FOR EXAMPLE A Cannae Drive uses an 805 MHz TM010 mode cavity for accelerating a 5000 kg space vehicle with total continuous thrust of 5 newtons. To create this thrust, the EM wave would require 3.86 x 10-21 joules per cycle or 3.1 x 10-12 watts of field energy to provide the 5 newtons of continuous thrust.  These numbers are derived by multiplying the single pulse energy depletion by the cavity frequency to get the power requirements for maintenance of the EM-wave energy . Additional energy is input to the EM wave to make up for ohmic losses on the cavity walls.

With every new pulse of the resonating EM wave, the Cannae Drive starts a new pulse of acceleration. The resonating EM wave thrusts the Cannae Drive for a period of 1/f seconds.  Then a new field is generated in the same inertial frame as the new inertial frame of the accelerated Cannae Drive. During the pulse period, power requirements to achieve the acceleration are very close to zero. This system of operation is analogous to starting a new journey with every cycle of the resonating EM wave. The first step of the journey is always the easiest, and the first step is the only step the Cannae Drive ever takes.

NOTE In the inertial frame of the Cannae Drive, energy is always conserved. For a Cannae Drive to move uniformly from a v0 frame to a v’ frame and then uniformly back to a v0 frame, equal amounts of energy will be used by the resonating EM wave in both the acceleration and deceleration phases (this is a Newtonian approximation since the relativistic mass of the vehicle will decrease because the potential energy source of the vehicle is depleted as kinetic energy of the system is changed).

#### CONSEQUENCES FOR SPACE TRAVEL

The generation of an unbalanced force can be used to create substantial improvements for space propulsion. With relatively low power inputs into a Cannae Drive, extremely high speeds, even approaching c, can be achieved. The massive amounts of propellant and energy required by reaction-mass-based propulsion systems are not required by the Cannae Drive. Further discussion on the advantages of the Cannae Drive are contained in the Applications section of this web site.