The consistency of Lorentz transformations in special relativity is ensured by the principle of relativity and the invariance of the speed of light.
The principle of relativity states that the laws of physics should be the same in all inertial reference frames. An inertial reference frame is a frame of reference in which an object or observer moves with a constant velocity or remains at rest. Lorentz transformations are mathematical equations that describe how physical quantities, such as time, space, and momentum, are related between different inertial reference frames moving relative to each other.
Lorentz transformations involve a combination of space and time coordinates and incorporate the concept of time dilation and length contraction. These transformations allow for the consistent description of physical phenomena in different frames of reference.
When you perform a Lorentz transformation to go from one reference frame to another, you apply the equations that account for the relative velocity between the two frames. These equations correctly account for the effects of time dilation and length contraction. They ensure that the fundamental laws of physics, such as conservation of energy and momentum, hold true in both reference frames.
Now, when you reverse the transformation, going back to the original frame of reference, you apply the inverse equations of the Lorentz transformation. These inverse equations mathematically undo the effects of the initial transformation, returning you to the original frame of reference.
The consistency of Lorentz transformations is maintained because they are derived from the fundamental principles of special relativity. These principles, such as the constancy of the speed of light in all inertial reference frames, provide the framework for the consistent behavior of physical quantities across different frames of reference.
In summary, Lorentz transformations stay consistent because they are based on the principles of relativity and the invariance of the speed of light. These transformations correctly account for the effects of time dilation and length contraction, ensuring that physical laws hold true in different inertial reference frames.