NEGATIVE MASS IN EINSTEIN'S ELEVATOR
by
Edmond S. Miksch
ed_miksch@yahoo.com
Copyright: September 1, 2007
Negative Mass in Einstein's Elevator
In the negative mass home page, www.Negative-Mass.com , we considered a gravitational field that is produced, according to the law of gravity, by massy particles. We showed that the field so produced has a negative mass density. By the principle of equivalence, we should then expect an acceleration field seen by an observer in an accelerating frame of reference to also have a negative mass density.
In one of his thought experiments, Albert Einstein considered an observer located in a spacious chest (or elevator) in empty space, far from appreciable masses. A rope is attached to the center of the lid of the chest, and an external being pulls on the rope. An observer inside the chest has no way of knowing whether he is in such an accelerating chest in empty space, or is in a gravitational field, as on the Earth. According to the theory of general relativity, there is no difference between a gravitational field due to massive bodies such as Earth, and such an acceleration field.
For our present work we will be considering very small differences in acceleration, and need to modify this statement slightly to say that there is no difference between a gravitational field wherein the field vectors are all parallel, and an acceleration field seen by a linearly accelerating observer. A gravitational field on the Earth having parallel field vectors could be provided in a small experimental volume by massy rings placed below and around the experimental volume.
The "principle of equivalence" states that the two fields are equivalent. We should, therefore, expect that an observer who accelerates relative to field-free space so as to experience an acceleration field, would see otherwise empty space to have a negative mass density.
We propose a thought experiment in which the chest or elevator is replaced by a rail car that accelerates on a track that is adjacent a platform. The platform serves as an inertial frame of reference for our calculation. The track and platform are not considered to be in any gravitational field. We visualize them floating in space, far from appreciable masses. We define coordinates, x,y,z and t for the inertial frame of reference. This arrangement is sketched in Figure EE1. The rails are denoted r, the rail car as c and the platform as p. A right handed coordinate system x,y and z is shown, z being directed from the figure toward the reader. A mark on the platform denotes x = 0. The coordinate x is parallel to the track. The end of the rail car having the greater value for x is referred to as the front end, and the end having the lesser value is referred to as the rear end. The rail car accelerates in direction x with a constant value, denoted a. More precisely, the front end of the rail car accelerates at the constant value, a. The length of the rail car at rest is denoted L0.
We consider movement of the rail car such that at t = 0, the front end of the rail car is at x = 0. Since the acceleration of the front end of the car is denoted a, the velocity of the front end of the rail car is v = a * t. The time t, seen by observers on the platform, may be negative or positive. We may consider, at times prior to time t = 0, the rail car to be disposed with its front end at a positive value of x, and its velocity to be negative. The acceleration a is then a braking acceleration which brings it to a stop at time t = 0, with its front end at x = 0. As time becomes positive, the continuining acceleration causes v to become positive, increasing as v = a * t, and the rail car begins moving toward the right.
A very brief statement of our method of calculation is this: We note that as the rail car accelerates and thus builds up velocity, it begins to show the effects of Lorentz contraction, as seen by observers on the platform. The observers on the platform will therefore see the rear end of the rail car to have a greater acceleration than the front end. If we then find that the acceleration of the rear end of the rail car is greater than the acceleration of the front end, as seen by observers in the rail car, then we have shown that a positive divergence of the acceleration field is seen by observers in the rail car, and this corresponds to a negative mass density. Note that no flux of the acceleration field vector enters or exits through the sides, top or bottom of the rail car. That is because coordinates y and z are not affected by a velocity in the x direction. Measuring rods oriented in the y and z directions neither shrink nor expand as a result of velocity in the x direction. (This follows from the Lorentz transformation.)
Viewing from the Platform
To obtain the acceleration of the rear end of the rail car, as seen by observers on the platform, we first obtain the time rate of change of L.
removing cancelling factors:
We differentiate Equation EE2 again to obtain the second derivative of length with respect to time. To employ the quotient rule for differentiation, we set:
hence,
The time derivatives of N and D are:
We are calculating the acceleration at the time t = 0. Hence, dD / dt = 0. Therefore,
At the time t = 0, the denominator D equals unity. Hence, we get:
Hence, still viewing the rail car from the perspective of observers on the platform, and denoting the difference between the acceleration of the rear end of the rail car and the acceleration of the front end of the rail car as Δa, we get:
Therefore:
To put this in more familiar notation, we note that the rail car itself serves only as a measuring rod which changes length as it accelerates. We accordingly denote the length of the rail car as ΔX, obtaining:
Viewing from inside the Rail Car
We now inquire as to what acceleration fileld would be seen by observers inside the rail car at the time T = 0, when the velocity of the rail car is zero. One would expect that a clock attached at the rear end of the rail car would at that instant be running at exactly the same rate as a clock on the platform adjacent the rear end of the rail car, since the velocity of the rail car (and its rear end) is zero at that instant. Likewise, One would expect that a clock attached at the front end of the rail car would at that instant be running at exactly the same rate as a clock on the platform adjacent the front end of the rail car, since the velocity of the rail car (and its front end) is zero at that instant. Likewise measuring rods on the rail car at the two ends would exactly agree with adjacent measuring rods on the platform. Therefore, observers at each end of the rail car would agree eactly with adjacent observers on the platform regarding the accelerations of the ends of the rail car. Thus, the acceleration field seen by observers inside the rail car would be that given by Equation EE5.
Observers inside the rail car would, therefore, see an acceleration field having the following gradient in the X-direction:
Thus, the divergence of the acceleration field vector, as seen by observers inside the rail car is:
Because of the principle of equivalence, we compare this value with the corresponding formula for the divergence of the gravitational field vector due to the negative mass density of the gravitational field. We obtained this in the negative-mass.com home page, in Equation G5. Equation G5 is here rewritten to employ the vector a rather than g, which we can do because an acceleration field is believed to be completely equivalent to a gravitational field.
We ought to see exact agreement between Equation EE7 and Equation G5 because of the principle of equivalence. The factor of 2 discrepancy must be due to an error in the preceeding calculation. The reader is challenged to find the mistake. We note that we have found the acceleration field seen by an observer in the rail car to have a negative mass density, but we obtain twice as great a negative mass density than we would expect.
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