Transformation

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60. The standing wave exists in a microwave resonator if the length of the resonator cavity is equal to multiple half-wavelengths of microwave. The stationary interference of standing wave will travel in another inertial reference frame. The vibrating pattern of the standing wave is conserved.

The existence of nodes in all reference frames requires the wavelength of the microwave to be conserved in all inertial reference frames. The angular frequency of microwave is different in every reference frame. Hence, the apparent velocity of the microwave depends on the choice of reference frame while the elapsed time remains invariant in all reference frames.

59. The total momentum of an isolated system is invariant if the system is subject to a conservative force. An isolated system of two identical objects subject to the gravitational force presents a rigorous proof that the mass of the object is independent of the direction of its motion.

In an inertial reference frame moving in the transverse direction to the direction of gravitational force, the law of conservation of momentum requires the mass of an object to be independent of its speed.

57. A standing wave can be formed in a microwave resonator if the length of the resonator cavity is equal to multiple half wavelengths. The stationary standing wave becomes a moving standing wave in another inertial reference frame.

The covariance property of the moving standing wave verifies that the frequencies of two microwaves forming the standing wave become different in the new reference frame while the wavelengths remain identical.

Hence, the apparent speed of the microwave appears to be different in a different inertial reference frame.

28. The conservation of mass in inertial reference frame is a property of the conservation law of momentum.

The inelastic collision between two identical objects shows that the total momentum is zero in the COM frame (center of mass). In another inertial reference frame, the rest frame of one object before collision, the total momentum before the collision is equal to the total momentum after the collision.

This conservation of momentum shows that the mass of an object is also conserved in all inertial reference frames. The mass of a moving object is independent of its velocity.

28. The application of symmetry to physics leads to conservation law and conserved quantity. For inertial reference frames, the reflection symmetry generates not only conservation but also transformation. Under reflection symmetry, the elapsed time is conserved in all inertial reference frames. The displacement in space is also conserved in all inertial reference frames. From the conservation of the elapsed time and the displacement, the coordinate transformation between inertial reference frame is derived. Based on the coordinate transformation, both the time transformation and the velocity transformation are also derived. The derivation shows that all three transformations are dependent exclusively on the relative motion between inertial reference frames.

10. An isolated physical system of elastic collision between two identical objects is chosen to manifest the conservation of momentum in two inertial reference frames. In the first reference frame, the center of mass (COM) is stationary. In the second reference frame, one object is at rest. The second frame is created by a temporary acceleration from the first frame. By applying both velocity transformation and conservation of momentum to this isolated system, mass transformation is derived precisely. The result shows that the mass of an object is independent of its motion.

9. Time in an inertial reference frame can be obtained from the definition of velocity in that inertial reference frame. Velocity depends on coordinate and time. Therefore, coordinate transformation and velocity transformation between inertial reference frames can lead to time transformation. Based on this approach, the time transformation between two arbitrary inertial reference frames in one dimensional space is derived. The result shows that the elapsed time is identical in all inertial reference frames.

8. A moving object in one inertial reference frame always moves at a different speed in another inertial reference frame. To determine this different speed, a temporary acceleration is applied to a duplicate of the first inertial reference frame in order to match the second inertial reference frame. The velocity transformation between two inertial reference frames is precisely derived based on the applied acceleration. The result shows that velocity transformation depends solely on the relative motion between inertial reference frames. Velocity transformation is independent of the speed of light.

6. Two inertial reference frames moving at identical velocity can be seperated if one of them is put under acceleration for a duration. The coordinates of both inertial reference frames are related by this acceleration and its duration. An immediate property of such coordinate transformation is the conservation of distance and length across reference frames. Therefore, the concept of length contraction from Special Relativity is impossible in reality and physics.

5. Two inertial reference frames moving at identical velocity can be seperated if one of them is put under acceleration for a duration. The coordinates of both inertial reference frames are related by the acceleration and its duration. An immediate property of this coordinate transformation is the conservation of distance and length across reference frames. Therefore, the concept of length contraction from Lorentz Transformation is impossible in reality and physics.

1. Translational symmetry in one-dimensional space requires the distance between two objects moving at equal speed under equal acceleration to be constant in time. However, motion between the object and the observer is relative. Therefore, this distance is constant in time for an accelerating observer. Consequently, the length of an accelerating object is constant in time. The length of an moving object in the direction of motion is independent of its speed.