#inertial force

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Relationship between vacuum resistance and momentum energy

Here I focus only on the magnitude of the vectors of force and momentum.

Force is generally defined as the time derivative of momentum.

where F is force, P is momentum, and τ is proper time.

Both sides of the above formula are multiplied by the speed of light in order to convert momentum to energy.

where c is the speed of light and Es is momentum energy.

Therefore, the formula of momentum energy is expressed as follows:

Momentum energy is obtained by multiplying the time integral of force by the speed of light.

When the direction of the force to be applied to a matter is the same as the direction of motion of the matter, the force is expressed by the following formula.

where R is the vacuum resistance, γ is the Lorentz factor, and v is the relative space speed of the matter with mass m.

Therefore, if the acceleration (dv/dt) is constant, as the space speed v increases, the Lorentz factor γ increases as shown in the figure below.

This means increased vacuum resistance. Hence the force required for the same acceleration increases, resulting in an increase in the momentum energy.

The amount of energy required for the same acceleration changes according to the characteristics of vacuum resistance.

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Difference between vacuum resistance and centrifugal force

When the direction of the force applied to a matter is different from the direction of motion of the matter, the force vector can be decomposed into components that are the same as and perpendicular to the direction of motion.

The space speed is increased by the same force component as the direction of motion or decreased by the force component opposite to the direction of motion. Vacuum resistance is generated by this linear force component.

The characteristic formula of vacuum resistance is as follows:

where γ is the Lorentz factor, m is mass and a is acceleration. The negative shows the opposite direction of force.

On the other hand, the direction of motion is changed by the force component perpendicular to the direction of motion. Centrifugal force is generated by this centripetal force component.

The characteristic formula of centrifugal force is expressed in the following formula.

The centripetal acceleration in Newton's laws of motion can be approximately expressed in the following formula.

where v is the linear speed along a curved path and r is its instantaneous radius.

The approximate relativistic centrifugal force is obtained by substituting the centripetal acceleration formula into the centrifugal force formula as follows:

where c is the speed of light.

As rotational speed approaches the speed of light, centrifugal force becomes infinitely large  because the Lorentz factor  becomes infinitely large.

The black hole at the center of the Milky Way galaxy rotates at a speed close to the speed of light and generates centrifugal forces to withstand its own gravity.

The magnitude of vacuum resistance is always equal to that of the same force component as the direction of motion. On the other hand, we can actively change the magnitude of centrifugal force by changing rotational speed.

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Relativistic formula of motion

In order to accelerate a matter, a force that is equal in magnitude and opposite in direction to the vacuum resistance generated by accelerating it must be applied to it.

When the direction of a force is the same as the direction of motion of a matter, the force F is expressed by the following formula from the characteristic formula of vacuum resistance.

where R is the vacuum resistance, γ is the Lorentz factor, m is the mass, a is the acceleration, v is the space speed, and c is the speed of light.

This is the relativistic formula of motion.

As a result of the force applied to the matter, its space speed increases at the acceleration:

If the same force continues to be applied to the matter, the Lorentz factor is approximately 1 then its acceleration is approximately constant while its space speed is much smaller than the speed of light. However, the Lorentz factor becomes extremely large then the acceleration approaches zero as the space speed approaches the speed of light. This is the reason why protons or electrons hardly accelerate even if they are energized by an accelerator when their speed approaches the speed of light.

When light quanta without mass (m=0) are emitted, they accelerate at infinite acceleration to the speed of light which is the maximum speed in the universe. In other words, light quanta are emitted at the speed of light.

The relativistic formula of motion is approximately the same as the Newtonian formula of motion when the space speed is much smaller than the speed of light.

In the case of the Newtonian formula of motion, the space speed of a matter will eventually exceed the speed of light if a force continues to be applied to it. In contrast, in the case of the relativistic formula of motion, the space speed never exceeds the speed of light because the acceleration approaches zero as the space speed approaches the speed of light.

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Characteristic formula of vacuum resistance

Suppose that the matter with mass m and an observer are in the same inertial frame of reference and the matter moves at speed v relative to the observer.

Elapsed time is proportional to time speed, and the speeds of coordinate time t and proper time τ are c and √(c^2−v^2) respectively where c is the speed of light.

Hence the following equation is satisfied.

where γ is the Lorentz factor, and

Physical quantities of matter must be calculated using proper time.

The relationship between the space speed Vs measured by using proper time and the space speed v measured by using coordinate time is as follows:

where l is the distance that the matter moves.

Therefore, the magnitude of its momentum is expressed in the lower formula.

This is the formula that expresses the momentum measured by using proper time in coordinate time.

Let's assume that the direction of a force is the same as the direction of motion of the matter.

Force is generally defined as the time derivative of momentum.

Therefore, the vacuum resistance R is expressed by the following formula.

The negative shows that vacuum resistance is directed opposite to force.

Because dv/dt is acceleration expressed in "a", the above formula can be rewritten as:

This is the characteristic formula of vacuum resistance. It can be said that the Lorenz factor itself represents the characteristic of vacuum resistance.

Vacuum resistance is zero when mass is zero or a matter moves at a constant speed.

It can be seen that the vacuum resistance generated when accelerating becomes extremely large as the speed of a matter approaches the speed of light.

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Substance of inertial force

When a force is applied to a matter and it accelerates, a "vacuum resistance" that has the same magnitude and opposite direction acts on it.

Vacuum resistance is a reaction force that occurs only when a matter is accelerating. The substance of inertial force is vacuum resistance. Inertial force is an actual force acting on a matter, not a fictitious force.

The vacuum resistance is approximately proportional to the acceleration while the space speed of a matter is much smaller than the speed of light. However, slight acceleration generates large vacuum resistance when the space speed approaches the speed of light, which is the maximum speed in the universe.

Electrons hardly accelerate even if they are supplied with a large amount of energy when their speed approaches the speed of light. The reason is that the vacuum resistance acting on them is extremely large.

The source of vacuum resistance is the vacuum energy.

[stevenloves]

curvature + motion = inertial force