Relativistic time dilation

[Ignat]

Relativistic time dilation usually refers to the kinematic effect of the special theory of relativity, which consists in the fact that all physical processes in a moving body are slower than in a stationary body relative to a stationary (laboratory) frame of reference. Relativistic time dilation is manifested, for example, by observing short-lived elementary particles formed in the upper atmosphere under the influence of cosmic rays and managing to reach the Earth's surface due to it.
A quantitative description of time dilation can be obtained from Lorentz transformations.:

where Δt is the time passing between two events of a moving object from the point of view of a stationary observer, Δt0 is the time passing between two events of a moving object from the point of view of an observer associated with a moving object, v is the relative velocity of the object, c is the speed of light in a vacuum.

The experiments referred to in the proof of the Lorentz formula, in fact, do not prove anything. Indeed, consider, for example, an increase in the lifetime of muons when they are accelerated to the speed of light. It follows from the theory of relativity that processes in a reference frame do not depend on whether it moves uniformly at a certain speed or is at rest. Moreover, due to the invariance of reference frames, observations from either of the two systems moving relative to each other are equivalent. If the Earth is considered a fixed point, then when the muon moves at the speed of light, according to the Special theory of relativity, the processes inside it will slow down and its lifetime will increase, which is supposedly what the experiment shows. But from the point of view of the same Special theory of relativity, if the observer is on the muon, then the Earth is moving towards him at the speed of light and the Earth's lifetime should also slow down. Thus, time dilation will be the same in different systems, i.e. we can assume that it will not exist at all. Consequently, the muon lifetime increases not because it is traveling at near-light speed, but because, for example, with what acceleration the muon is gaining near-light speed. We see that this experiment does not confirm the Lorentz formula in any way. It follows from this that the Lorentz formula does not describe real processes, but is a pure abstraction, or perhaps it is simply not true.

If we move on to heavier macro objects, then when they accelerate to near-light speed, which in itself is fantastic, they will inevitably collapse and it is not possible to verify the correctness of the Lorentz formula. Measuring time using periodic signals (atomic clocks) and their lag in moving objects is not evidence of a slowdown in physical processes in a moving system, but only indicates that the frequency of the oscillatory system is changing.

The same story happens when considering the Havele-Keating experiment. Already looking at the simplified circuit of an atomic clock, it becomes clear that it must be protected from external factors, otherwise the measurement error will be high.

 

[Unclear Engineer]

But from the point of view of the same Special theory of relativity, if the observer is on the muon, then the Earth is moving towards him at the speed of light and the Earth's lifetime should also slow down. Thus, time dilation will be the same in different systems, i.e. we can assume that it will not exist at all.

That part where I think you are missing something important.

If I am on a muon heading towards Earth at nearly the speed of light, and the processes of whatever is happening on Earth appear to be slower to me, then a muon created at the same time that the muon I am riding on should not decay before I get there on my muon, right?

The problem with that is the two frames of reference will not agree on what the same time is in different places. Simultaneity in relativistic situations requires the same time and place to be immutable between frames of reference.

An apparent paradox used in teaching Special Relativity involves a guy with a ladder that is longer than a shed. He takes the ladder and gets a running start at the shed with the near door open and the far door closed. Maxwell's demons are assigned to close the first door as soon as the ladder is completely past, and open the far door as soon as needed to prevent it from hitting that door. The observer standing by the shed and watching this sees the ladder get shorter due to the high velocity, so it looks like it fits in the shed and he sees the near door close before the far door opens. But, the guy running with the ladder does not see the ladder get shorter, he sees the barn getting even shorter than it looked before he started moving so fast. So, to him, the far door needs to open even sooner than the near door needs to close than he calculated before he started running (because he did not use the Lorentz transformations to do that calc).

Try adding the -vx/c term to your calcs and see what you can do with that complete equation.

EDIT: Not sure what happened to my equation paste - it looks OK in edit mode, and I thought it looked OK when first posted. But, now there is a red "X" in its place. Maybe I violated some source prohibition and the moderators blocked it?

Anyway, the equation is t' = gamma x (t - vx/c^2). It is the -vx/c^2 term that needs to be taken into account by the OP.

 

[AJ Trahan]

I've been exploring an alternative way of looking at time dilation that treats it as a physical field effect, not just a coordinate artifact. The idea is that time flows as a real field in the universe, with a characteristic speed (like the speed of light in vacuum). When an object gains kinetic energy, it compresses the local "temporal field" around it. This compression physically slows all internal processes — including particle decays and clock rates. The usual Lorentz factor still applies mathematically, but here it represents actual field compression, not a passive observation. This interpretation also cleanly resolves the reciprocity issue — only the object with added energy compresses its local temporal field. The approach is fully compatible with the experimental results we observe (muon decay, Hafele–Keating, GPS corrections), but provides a more intuitive, causal picture of what’s happening.

Gamma_T = T_ambient / T_local = 1 / sqrt(1 - v^2 / c^2)

 

[Unclear Engineer]

Interesting. Relativity Theory has been a good description of reality, but not really an explanation. Attempts at explanation talking about space bending and flowing don't seem to get both support from theorists and understanding by others at the same time.

I wonder how your field theory would work out for the time dilation by proximity to mass in General Relativity. I long ago noticed that the time dilation at the surface of a mass is the same as the Special Relativity velocity time dilation for an object at that location moving at the escape velocity from that location, compared to an observer at infinite distance and zero relative velocity to that mass. That seems to be consistent with "space" flowing into the mass at escape velocity. But, "flowing space" seems to be a red line that theorists fear to cross.

So, how does your field theory look for that situation?

And, if that works out for you, what does your theory show for time dilation beneath the surface of the mass, as the object approaches the center of mass? That is an experiment that I would like to see actually performed physically, but it is actually pretty difficult to do, physically. As I see it, the options are that the time dilation continues to track the escape velocity, which continues to increase with depth into the mass, or it tracks the net gravitational force on the mass, which goes to zero at the center of mass. The result of a physical experiment would test whether the time dilation effect is a function of the force or simply the proximity to mass. I suspect your field equation would be the second case.

 

[AJ Trahan]

Very sharp observations — I’ve come to similar conclusions. The equivalence with escape velocity at the surface is not accidental; it reflects the local gradient of time flow. Treating time itself as an active field (rather than invoking "flowing space") resolves this neatly.


As for the interior: In the model I’m working on (Unified Temporal Gradient Theory), time dilation tracks proximity to total enclosed mass, not local force. At the center of a symmetric mass, even though gravitational force drops to zero, time dilation reaches its maximum — because you are embedded within the full mass effect.


It would be a fantastic experiment to perform, and this is one of the clean predictions such a model offers. If you'd like, I can share the exact field equation — it's very illuminating when you run it for interior regions of mass

 

[AJ Trahan]

Here’s the core idea: in the UTG framework, the temporal field T(r) inside a mass is computed based on the cumulative mass enclosed at each radius. I can show you the equation — it makes very clear why time dilation continues increasing toward the center, independently of force.

For a uniform spherical mass of radius R and total mass M, the time dilation factor at radius r inside the mass is:

T(r) = sqrt(1 - (2G * M(r)) / (r * c^2))

where M(r) = M * (r^3 / R^3) is the mass enclosed within radius r.

You can see that as r → 0, M(r) → 0 faster than r → 0, but the total potential accumulates — leading to maximum time dilation at the center, even though gravitational force is zero there.

It’s a clean and testable prediction — very different from what you’d expect if time dilation simply tracked gravitational force.

 

[Unclear Engineer]

Interesting, but I don't have the time right now to play with the math. I am assuming that the total time dilation at a point requires integration over r for M(r), so that on the surface of a large sphere with radius R and total mass M, the integration would need to go out to 2*r to include the entire sphere.

So, let me ask:

1. Does your equation produce the same time dilation at the surface of the large mass M with radius R as we get with GRT?

2. As the point of observation travels from the surface of the mass M with radius R down to the center of the large mass, does your time dilation track with the escape velocity from r (out to infinity). or is it a different function of r?

 

[AJ Trahan]

Great questions — I’ll clarify:

Surface time dilation: Yes, the UTG equation produces the exact same result as GRT at the surface of the mass — matching the Schwarzschild solution:

T_surface = sqrt(1 - 2GM / (R c²))

This is fully consistent; UTG is constructed to match established GRT predictions where those are already verified.

Inside the mass — does it track escape velocity or something else?

Here’s where UTG diverges conceptually:

• Time dilation does not simply track escape velocity from r → ∞.
• Nor does it track local gravitational force.

It tracks the total integrated temporal potential due to the enclosed mass.
In other words, at radius r inside the sphere, the time dilation is governed by:

T(r) = sqrt(1 - 2G * M(r) / (r c²))

where M(r) = M * (r³ / R³).

Now — escape velocity from radius r also depends on M(r), but through a different relationship:

v_esc(r) = sqrt(2G * M(r) / r)

At the surface, both expressions align (hence your observation about surface matching escape velocity). But inside the mass, they diverge:

Escape velocity keeps increasing as you move inward.

• UTG time dilation increases toward the center based on cumulative enclosed mass, not by tracking the escape velocity curve.

Summary:
Surface time dilation — same as GRT.
Interior time dilation — tracks cumulative mass effect, not escape velocity per se.
At center of mass: maximal time dilation, gravitational force zero, escape velocity finite but not the governing factor for T(r).

If you’d like, I can provide the exact expression for the total integrated time dilation from r → ∞, and show how it compares at different depths. It’s a very revealing comparison.

 

[AJ Trahan]

As a side note — this is part of a full formal framework I’ve been developing (Unified Temporal Gradient Theory), which is currently under peer review. It addresses this and several other open questions in gravitational and quantum physics. If you’d like, I can share the current version.

 

[Unclear Engineer]

I would like to see the equation, but those are hard to produce here. Maybe a graph of the results of your equation and GRT as a function of r from 0 to at least R? That would also probably have to be a URL.

Edit: Yes, I would like to see your current draft. Do you have a link to it, or is it just in pdf?

 

[ChrisA]

Interesting. Relativity Theory has been a good description of reality, but not really an explanation. Attempts at explanation talking about space bending and flowing don't seem to get both support from theorists and understanding by others at the same time.

I wonder how your field theory would work out for the time dilation by proximity to mass in General Relativity. I long ago noticed that the time dilation at the surface of a mass is the same as the Special Relativity velocity time dilation for an object at that location moving at the escape velocity from that location, compared to an observer at infinite distance and zero relative velocity to that mass. That seems to be consistent with "space" flowing into the mass at escape velocity. But, "flowing space" seems to be a red line that theorists fear to cross.

So, how does your field theory look for that situation?

And, if that works out for you, what does your theory show for time dilation beneath the surface of the mass, as the object approaches the center of mass? That is an experiment that I would like to see actually performed physically, but it is actually pretty difficult to do, physically. As I see it, the options are that the time dilation continues to track the escape velocity, which continues to increase with depth into the mass, or it tracks the net gravitational force on the mass, which goes to zero at the center of mass. The result of a physical experiment would test whether the time dilation effect is a function of the force or simply the proximity to mass. I suspect your field equation would be the second case.

Proximity to mass vs. gravity field. This is an easy experiment. You do not need to go to the center of the Earth. All you need to do is be closer to the center. A deep coal mine could work. And if you want to test this in space, a Lagrange point works.

I suggest not bothering to try the experiment because you will not be the first. It turns out that Einstein was in fact correct,

New clocks are quite good at measuring gravity. The difference between the gravity field at desktop height and on top of a bookshelf is measurable with today's clock technology. It turns out that it is in fact acceleration that affects time.

 

[AJ Trahan]

Thanks — I really appreciate your interest. I haven’t generated a graph yet, but the comparison is straightforward using the field equation I shared. The UTG prediction inside the mass tracks cumulative mass (not escape velocity), and aligns with GRT at the surface.

As for the full theory — yes, I have the complete draft available. It’s written as a formal framework (RTF format), covering the core field equations, cosmological implications, quantum behavior, and testable predictions. Unfortunately I'm unable to post the web address here and being new to this I have no idea how to PM or anything like that.

 

[AJ Trahan]

Proximity to mass vs. gravity field. This is an easy experiment. You do not need to go to the center of the Earth. All you need to do is be closer to the center. A deep coal mine could work. And if you want to test this in space, a Lagrange point works.

I suggest not bothering to try the experiment because you will not be the first. It turns out that Einstein was in fact correct,

New clocks are quite good at measuring gravity. The difference between the gravity field at desktop height and on top of a bookshelf is measurable with today's clock technology. It turns out that it is in fact acceleration that affects time.

 

[AJ Trahan]

Good points — and you're absolutely right that modern clocks can measure these effects with remarkable precision.


One small but important clarification: in both General Relativity and the framework I’m working on (Unified Temporal Gradient Theory), it’s not the local gravitational acceleration that governs time dilation — it’s the gravitational potential, which integrates the mass distribution.


That’s why time dilation can be maximal at the center of a mass where gravitational force (acceleration) is zero.


In the UTG model, this is treated explicitly as a function of cumulative proximity to mass, rather than force. The difference is subtle but has important implications — especially in interior regions and near complex mass distributions.

 

[Unclear Engineer]

Chris, please post a link to any experiment that shows a decrease in acceleration due to gravity and a correlated decrease in the time dilation effect as a function of depth inside a mass.

Perhaps a deep coal mine could do that, but I suspect there would be a lot of corrections for temperatures, uneven mass distributions near the surface, etc. So, it would need to be measurement of both effects together in the same experiment as a function of depth to really "seal the deal".

 

[Ignat]

The idea is that time flows as a real field in the universe, with a characteristic speed (like the speed of light in vacuum).

Interesting reasoning, but how can time flow if it's not physically there? Time is just a tool with which we can judge the speed of processes in the material world.

 

[Unclear Engineer]

Time seems to be one of the things that we least understand.

There are people who think it is a fundamental property of the universe, and others who think it is an "emergent" phenomenon - something we perceive as a result of other things.

It is definitely a hard subject for a human to wrap his/her head around, especially when it comes to relativistic situations.

So, I try to keep an open mind and look at how other people are trying to model it.

 

[Classical Motion]

A vertical rotating shaft shows that time does not change with elevation.

A rotating shaft used as a swinging radius shows that neither velocity or acceleration changes time.

Only our present clocks(rate ticker) vary, with g and acceleration, not time.

A rotational clock ticker, incident to a force or a potential, instead of an oscillation ticker, will not vary.

Hayseed physics model.