# Relativity and variable time - there is an alternate formulation where time is always linear. Has it been studied?

Lorentz, Einstein et. al. assume time is the variable which changes in a gravity well or as speed approaches $$c$$. That's the commonly accepted model.

For nearly 50 years I've wondered if anyone else has considered introducing a hypothetical, "universal" rate of reaction U. Wherever t appears in all physical, chemical or electromagnetic formulae, it would be replaced with (U t). The product (U t) behaves precisely as t alone. However, in a case where t slows to 1/2, in this approach t remains the same, but U decreases by 1/2.

For an observer moving close to light speed, their chemical and physical reactions are slowed, including their atomic clock which measures time in their reference frame. From their perspective, the effect is identical to time itself slowing - but time is still progressing as it always has. Only their relative measurement of time has changed.

Since it's only a variable substitution, every formula in the cannon is unchanged. The only difference is that time is linear and universal - with U varying to compensate for relativity.

There is no observable, physical reason to believe in the U factor, but it makes the time ordinate linear and should make all the math easier.

Now, when doing an integral of time in an accelerating reference frame, it's possible to do a simple integration in t without t varying while it's being integrated.

I haven't taken the time to dig into this deeply, but it appears that mass itself might not be changing while its inertial changes because momentum = m v = m/U dx/dt. By factoring U as part of m, them m = m/U and once again, the rate of reaction changed - not the actual mass.

I asked this ages ago in physics classes and the answer was basically "we don't do it that way." That wasn't very satisfying.

Has anyone ever looked at this approach?

• Huh? Coordinate time is mainstream physics, and very useful. Commented Mar 6 at 21:05
• Why was this disliked? Clearly the OP has thought about the topic but doesn't come from a physics background. Commented Mar 7 at 13:22
• You can measure time in any units you like, as long as you keep track of those units. But you shouldn't expect an arbitrary choice of units to tell you anything interesting about how the world works. Commented Mar 7 at 16:11
• @BobT : any formulation of time you create will have to come to grips with the relativity of simultaneity, which means that no uniform definition of time is possible that applies to all observers. It also means that any mapping between users' coordinate times will necessarily involve not only time, but also space, because a "moving" observers clocks will be found by the "stationary" observer to be out of phase with one another. Commented Mar 7 at 16:32
• @EricSmith The way we do that with GPS is that the process locates the receiver in spacetime in the GPS coordinate system. That makes a uniform definition of time possible for all GPS users, regardless of their state of motion: they must simply accept the authority of GPS over their local clocks. That, in turn, could lead to conceiving of local processes as being subject to the kind of effects the OP postulates. And while that's not the way we teach the theory, sometimes it's how we apply it, "correcting" clocks for altitude, for example. Nothing wrong with it. Commented Mar 7 at 17:31

That's essentially what the Lorentz factor is. It's a unitless constant which scales the measured time or length in one frame to give the measured time or length in another frame. The important thing to notice is that "time slowing down to 1/2" is still dependent on a given reference frame and there's nothing universal about it. Additionally, time is always linear for a given frame. It's more so about how a reference frame will measure changes in time for other frames. But this wouldn't even be a problem since special relativity only concerns inertial frames (i.e., frames of constant velocity). So there's still not much of a benefit to the math. The Lorentz factor is still an important idea though and is one of the things which helped convince other physicists of Einstein's work.

Nothing prevents you from choosing a reference frame and labeling events in spacetime according to that frame. That's essentially how GPS works: GPS time isn't controlled by the clocks on the satellites, but by a notional clock at sea level. What relativity tells you is that this choice is arbitrary.

• Yes, that's correct. But the Lorentz/Einstein assumptions are that mass and time change. My question was, has anyone ever studied an alternate derivation where time, mass and distance remain unchanged but uses another variable, which I refer to as rate of reaction, to explain the effect? This is basically, a yes/no question. Have they or haven't they? Your response is along the "that's the way it's done" approach and I'm wondering whether another way was ever studied?
– BobT
Commented Mar 7 at 3:18
• @BobT Whether mass changes depends on how you define mass. The current fashion is to define it in a way that it doesn't change between reference frames ("invariant mass"). So that's most certainly been studied. What you call "rate of reaction" is a product of time dilation and gravitational redshift. That's there in our calculations. We don't call it "rate of reaction", but the Universe doesn't care what words you use. Commented Mar 7 at 3:39

Let me try to translate your thoughts into main stream physics words and then you can see if I understood you correctly.

You are searching for a variable, call it $$t$$, that has two properties:

1. This variable is reference frame independent (coordinate free)
2. There exists a linear transformation (call it $$U$$) that is dependent on the velocity with respect to which someone moves relative to the person that defines the laws of the universe with $$U=1$$. With this linear transformation one can relate your time to any other time.

Fortunately, this is a known quantity called proper time. It is an invariant quantity. For each instance in "time", we can calculate the $$U$$ needed for other people to translate whatever their notion of time is to ours, thus recovering what we understand as laws of physics.

Read up proper time to get a better understanding. It is indeed the thing you are asking for. (notice that in the "proper frame" I am sitting in the coordinate 0.)