Relativistic mass distortion

Relativistic Mass Distortion

Part I: Mass Distortion by Velocity

It has long been known that as an object approaches the speed of light (c), its mass increases without bound. If a moving object passes by you (a stationary observer) at a substantial fraction of a speed of light, then the mass of the moving object is...

m'=γm

m' is the mass of the object as seen by the stationary observer

m is the inertial mass of the object

γ is the Lorentz factor

u is the velocity of the moving object

c is the speed of light in a vacuum

This can be proved by using a special type of scale that can be used to measure an object’s mass in the absence of gravity. This scale can be made by placing a mass between two springs that are attached to fixed supports. The longer the period of oscillation (of the springs and the mass), the higher the mass. Now, lets say that the period of oscillation for a mass in this scale is 1 second as measured by the observer of a spaceship moving at 0.9c. To a stationary person outside the spaceship, one second to a person on the spaceship is now roughly 2.29 seconds, so the period of oscillation is now 2.29 seconds, and therefore the mass on the scale is 2.29 times its rest mass. This rule must also include the symmetrical effect of time dilation at a constant velocity, and therefore the person on this spaceship will see the stationary observer as having 2.29 times his rest mass.

Part II: Mass Distortion by Rotation

Mass can also be distorted by rotational motion. This effect is caused by the tangential velocity of a mass that is rotating around a central axis. This, like linear mass distortion, can be seen by using the spring scale, whose period is affected by time dilation.

ω=θ/t

d=θr

d=distance(arc length) traveled

r=radius

v(tangential)θr/tωr

therefore....

γ=1/√(1-ω r /c )

So, for an amount of mass that is rotating around a given point, the new mass is equal to....

m'γmm/√(1-ω r /c )

However, since a rotating reference frame is non-inertial, the symmetry of time dilation doesn’t exist. Therefore....

m'=m√(1-ω r /c )

m' is the mass of an object at the axis of rotation as seen by the observer rotating around that point

m is the mass of an object at the axis of rotation
the other variables are as listed above

Because mass and energy are equivalent, we can see that rotational time dilation can also affect energy.

E'=(mc )/√(1-ω r /c )

E' is the amount of energy in the mass of the object being rotated
m is the mass of the object at that point
We can also see that effect when the observer being rotated views a mass at the axis of rotation

E'=(mc )√(1-ω r /c )

E' is the energy of the mass at the axis of rotation that is being viewed by the rotating observer
m is the mass of the object at the axis of rotation

Part III: Mass Distortion by Gravity (Schwarzschild solution)

Because time dilation is an effect of gravity as well as an effect of rotation and velocity, then mass should also be affected by gravity. Using the spring scales to measure two equivalent masses with one in a gravitational field and one in deep space (no gravity), then the scale in the lower gravitational potential would appear to be measuring a larger mass than the other scale to a distant observer who is not in the gravitational field. The equation that describes this is

m'=(m/√(1-2GM/(rc )))-(Σ/c )

Σ is the gravitational potential energy that the object would have at the observer’s position minus the gravitational potential energy at its current position.

m is the mass of the object in the gravitational field

c is the speed of light

M is the mass of the object generating the gravitational field

r is a Schwarzschild coordinate describing the radial distance of the object from the center of mass of the massive object

m' is the mass as seen by a distant observer that is not in the gravitational field

Another important point is that a reference frame in a gravitational field is not an inertial reference frame, and therefore a distant clock (not in the gravitational field) appears to run faster to a person in a gravitational field. Therefore...

m'=(m√(1-2GM/(rc )))+(ψ/c )

ψ is the gravitational potential energy that the object has minus the gravitational potential energy that it would have at the observer’s position.

m' is the mass of the distant observer as seen by the observer in the gravitational field

m is the mass of the distant observer

c is the speed of light

M is the mass of the object generating the gravitational field

r is a Schwarzschild coordinate describing the radial distance of the observer in the gravitational field from the center of mass of the massive object
Because of the equivalence of mass and energy, we can also see that gravity can affect the energy of an object

E'c (m/√(1-2GM/(rc ))-(Σ/c ))m'c

E' is the energy as seen by a distant observer
the rest of the variables are as listed under the first equation

E'=c (m√(1-2GM/(rc ))+(ψ/c ))=m'c

E' is the energy of the distant observer as seen by the observer in the gravitational field
the rest of the variables are as listed under the second equation

Part IV: Violation of Conservation of Energy?

When it is seen that an object in a gravitational field appears to have more mass than it would outside of a gravitational field (to an observer outside of a gravitational field), it may seem that the law of conservation of energy may be violated (using the entire universe as a closed system). The conservation law is not violated at all. This is because spacetime is being “suckedï¿½? towards the object. Since this spacetime is actually moving, then the coordinate system is moving, and therefore the object that is stationary relative to the observer is actually moving at the velocity of the spacetime in the opposite direction of the moving spacetime. This also explains why the equations in the above section are ill-behaved at the event horizon and in the interior of the black hole: the equations assume zero velocity relative to the observer outside the gravitational field, but in being stationary they may exceed the speed of light relative to the spacetime.

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