Can Lorentz transformations hold only One-way from a preferred frame?

Question

I there any problem with assuming that Lorentz transformations hold only One-way from a preferred frame, let’s say stationary with respect to CMB. Still speed of light is constant and laws of physics are the same in every inertial frame, only the clocks in CMB are fastest comparing to the clocks in other frames. Is there an experiment that shows this cannot be the case or is there a theoretical contradiction?

Answer 1

I there any problem with assuming that Lorentz transformations hold only One-way from a preferred frame

Yes, there is a rather large problem: the math contradicts that assumption.

From the math of the Lorentz transform it is just a little algebra to get the inverse transform. The inverse transform is also a Lorentz transform. So the Lorentz transform unambiguously holds both ways.

Answer 2

When you include the CMB, you're dealing with general relativity and something like the FLRW metric. It's not special relativity, in which there is simply no way, ever, to define a preferred rest frame.

Many have looked at the CMB and claimed it defines a preferred rest frame, and it does. At a point. The problem is, that rest frame depends on position.

For instance, we can define our CMB relative rest frame here on Earth, but over at Andromeda (0.9 mega parsecs away), "their" rest frame is moving at 66 km/s relative to us.

So: no universal restframe.

Answer 3

As Dale has said, the Lorentz transforms are essentially two-way. I wanted to make a further point, which is that you must misunderstand Special Relativity if you believe that clocks in one inertial frame run faster than clocks in another.

EDIT- to address the points you have made in comments, you must misunderstand LET too, since you have stated in your question that clocks 'are fastest' in the preferred frame, and in your comment on JEB's question you say that 'clocks slow down' if they are moving relative to the preferred frame, and in your comments on Dale's answer you mention clocks slowing down if they move relative to the preferred frame. Your ideas are fundamentally flawed. In SR and in LET all good clocks tick off time at a rate of one second per second. What causes the time dilation effect is not that clocks are impaired in some way by their motion so that they under-report the passage of time- what causes it is that time in one frame is systematically out of synch with time in another. That is true both in SR and in LET. The lack of synchronisation is inherently two way.

So let us suppose there was an absolute preferred frame, and Alice is sitting in it at rest, and the LET applies. Now assume Alice is passed in turn by Bob and Carol who are coasting through the preferred frame at some speed. If Alice compares her watch firstly with the time on Bob's clock and secondly with the time on Carol's, she will see that the time difference between the two encounters is longer, according to the clocks of Bob and Carol, than it is according to her own watch. IE, her watch is time dilated and appears to be running slow, even though she is sitting at rest in the preferred frame.