We study the effect of the generation of the mechanical transverse wave (MTW)　travelling in the opposite direction (OD) to a moving medium (MM) on the relativistic energy conservation law (RECL). From the viewpoint of the relativity of simultaneity (RS), the time on the coordinate coinciding with the advance end of the wave (AEW) travelling toward the rear of the MM passes faster than that on the coordinate coinciding with the wave source (WS). Then the AEW in the MM travels forward compared to that in the rest frame of reference (RFR) which is stationary relative to the medium when the time on the coordinate coinciding with the WS is same for each inertial frame of reference (IFR). Hence, the coordinate interval (CI) between the AEW and WS in the MM is observed to be larger than that between them in the RFR. We show that this difference holds true for the CI of any portion having transverse velocities mutually converted by the Lorentz transformation (LT). This difference in the CI leads to that in the rest mass (RM). We demonstrate that the RM included in wave motion (WM) in the MM is larger than one included in WM in the RFR when comparing the portions having transverse velocities mutually converted by the LT. This relation holds true for all portions in WM. Therefore, the total coordinate interval of the portion (CIP) and total RM (TRM) included in WM in the MM (WMMM) are large compared to them included in WM in the RFR. Furthermore, we compare the relativistic kinetic energy (RKE) of the MTW travelling in the OD to the MM (ODMM) with that of the MTW propagating in the direction vertical to the moving direction of the medium. We prove that the CIP and RM included in the former MTW are larger than them included in the latter MTW when comparing each portion with the same transverse velocity (TV). Moreover, the total CIP and TRM included in the former MTW are also large compared to them included in the latter MTW. The reason for these is that the latter CIP and RM are equal to them in the RFR when comparing the portions having transverse velocities mutually converted by the LT. On the other hand, the energy supplied to generate each MTW is the same. From these, we demonstrate that the RKE of the MTW travelling in the ODMM can be larger than the total relativistic energy (TRE) of the MTW propagating in the direction vertical to the moving direction of the medium. Consequently, we propose a violation of the RECL and Einstein’s principle of relativity (EPR) because the TRE is not necessarily conserved in the IFR in which the medium is moving.

We study the center of energy (CE) before and after the separation of superposed wave from a moving medium (MM). It is assumed that two out-of-phase mechanical transverse waves propagating from the opposite directions on a medium moving at non-relativistic speeds are superposed and the superposed portion (SP) is separated from the MM at that moment. We consider the CE of the SP before and after the separation from the MM. The location of CE (LCE) of the SP seems to be at the center of it at the moment of superposition. The SP rotates due to the separation from the MM since the velocity of each portion symmetric with respect to the center of the SP is equal in magnitude and opposite in direction. The magnitudes of velocities of the symmetric portions become different as soon as the SP begins to rotate with the separation from the MM. Then the energies of their symmetric portions are not the same, so the LCE of the SP is not at the center of it. As a result, the LCE of it looks different before and after the separation from the MM. We must find a solution to keep the LCE of the SP constant. We propose that two out-of-phase mechanical transverse waves (MTWs) propagating from the opposite directions on a MM originally have hidden difference in relativistic energy and it suddenly appears within the range observable in Newtonian mechanics when the SP starts rotating. This means that the point of view of hidden difference in relativistic energy is necessary to keep the LCE constant.

We study whether the center of relativistic energy (CRE) is always constant in an isolated frame of reference (IFR). First, we assume that two objects in a moving isolated frame of reference (MIFR) move parallel to and very close to a coordinate axis from opposite directions and with equal speed. Here each length in the direction of travel of the objects is much longer than that perpendicular to the direction of travel of them, and moreover the latter is negligibly extremely short. Hence, the location of the CRE (LCRE) of the objects is very close to the coordinate axis in initial condition. When they become perfectly symmetric with respect to the coordinate axis, the forces perpendicular to the direction of travel of the objects are applied to each CRE and thereby a perfectly inelastic collision between them occurs on the coordinate axis. The combined object (CO) resulting from the perfectly inelastic collision begins to rotate because each momentum of the objects before the combination between them acts like the moment of force on the CO. For simplicity, we examine the energy distribution of the CO when it becomes perpendicular to the coordinate axis due rotation. The magnitude of velocity of each minute portion (MP) symmetrical with respect to the coordinate axis are different depending on whether the direction of each rotational velocity is the same or opposite to the direction of travel of the MIFR. Then the energy of each MP is not the same. As a result, the LCRE of the CO moves to the direction perpendicular to the coordinate axis, and its LCRE becomes far from the coordinate axis since the length in such direction is long enough. Therefore, we conclude that the LCRE in the MIFR is not necessarily invariant. Second, we suppose a process in which a force causes a negative acceleration on a moving object (MO) and thereafter the MO eventually comes to rest. What is considered here is the LCRE when the MO comes to rest. We pay attention to the position of the force applied to the MO because the transmission time of force (TTF) inside the MO may vary depending on the position of the MO where the force is applied. As a result, we find that the position at which the MO comes to rest differs depending on the TTF inside the MO. Therefore, we conclude that the LCRE in an IFR differs depending on the position of the MO where the force is applied, in other words, it in an IFR is not necessarily uniquely determined.