Strongest Force Ever Known: Time Difference
Chapter 5 from Classical Physics Beyond Einstein’s
As the book cover suggests (see section 3.2 for the details), time speed differs even at our head and feet levels: with each meter up, SECOND (time unit) becomes shorter/faster by 10⁻¹⁶ fraction of a second.
Even at the end of 20th century people were not able to measure such small difference in time speed: 1999 article “The Five Femtosecond Time Step Barrier” identifies 5×10⁻¹⁵ sec as “time step barrier” for science. That explains why neither we nor Newton have developed intuition of time flow variability. But nowadays physicists have tools and do experiments at 10⁻¹⁸ sec (which is called attosecond) timescale. And there are prospects for making clocks 1,000 times more precise, to work with 10⁻²¹ fraction of a second.
10⁻¹⁶ is a such a miniscule time difference, but it weighs a lot on us as gravity, which is “the weakest force in the Universe”. Or is it not? Time speed difference is denoted by a number D, it is called “time dilation” factor, by which time in one area runs slower than time in another area. Time speed varies even in the same area now and then: time in our Universe was twice slower (D=2) 14 billion years ago, and 12 times slower (D=12) 32 billion years ago (check Time Matters chapter 1). Let’s see now that there is no force stronger than a barrier between slow and fast time (even with D=2).
Chapter 5. CROSSING BORDER BETWEEN TIMEZONES
5.1. Mechanics by Snell’s Law: Escape Velocity
In chapter 2 we discussed that a light beam running in a slow time area, where time dilation is D>1, can penetrate into normal time area (where D=1) at an angle of incidence θ < arcsin (1/D) only:
Time Potential is 0.5×c²/D² in the area with time dilation D, and it is 0.5×c² in the area with D=1. An object of mass m, to get out from a slower time area needs extra energy m×(0.5×c² — 0.5×c²/D²). It can get it from its kinetic energy m×v²/2:
To cross the border, its velocity should be v > c×√(1–1/D²). Even more: perpendicular to the timezone’s border projection of this velocity needs such value, because parallel to the border velocity/projection does not participate in the border crossing.
The drawing above demonstrates that a particle with a velocity vector starting in the green cap area of the sphere of radius c with a center at the point O, can get through point O. Actually, we should have started velocity vectors at the object location near or at the point O, but then these vectors would span over the border, and that might be misunderstood. Escape (from D-time-dilated area) velocities start in the green cap:
5.2. Strong Force and Half-Life
Light, particles, objects cannot escape D=2 area at an incidence angle > 30° (because arcsin(1/D) = arcsin(1/2) = 30°):
In a spherical bead of twice slower time than outside, light and particles bounce off the surface at the same angle > 30° as the incidence angle, so they are trapped forever, unless something deflects them. From a non-spherical blob, light and particles can escape eventually at some point where incidence angle < 30°. Besides that, if we are talking about a particle, if its velocity < Cos(30°)×c = √3 / 2 ×c ≈ 0.86×c ≈ 259,807 km/sec, it can never escape from any D=2– blob, unless something boosts its velocity up. Time dilation, even at not very big D–numbers, is “strong force” capable of containing particles in a nucleus and photons in spherical particles for a long time, even indefinitely, if no severe disturbances/interactions. In particle and nuclear physics, the lifetime of particles and atomic elements is expressed by half–life: it is a time period during which half of the species (of these particles or nuclei) decay. Half–life is inversely related to probability of decay: longer lifetime corresponds to lower probability of decay in a fixed time period. Let’s explore what happens to a particle or a nucleus half–life, when time slows down inside them (time dilation factor D increases). Intuition suggests that lifetime of such species (which is inversely related to the probability of decay, and decay is about some parts escaping their confinement) is proportional to time dilation D, because from the outside perspective, velocities of parts inside the confinement decrease by D (see (1.1)), and so is frequency of escape attempts (number of contact points with timezone’s border, per outside–observer time unit) decreases. That is partially correct only, because probability of escape at a single contact with the border decreases by D⁴ as well, thus, combined with the reduction in escape attempts, decay probability decreases by D⁵. To prove that, let’s bring the last drawing from 5.1 for escape through point O velocities:
Let’s use a known formula for a spherical cap volume:
In our case, θ = arcsin (1/D): r = c, a = c/D, h = c×(1–sqrt(1–1/D²))
For small x, sqrt(1–x) ≈ 1–0.5x, thus: sqrt(1–1/D²) ≈ 1–0.5/D², and
h = c×(1–sqrt(1–1/D²)) ≈ 0.5×c/D²,
V ≈ 1/6×π×0.5×c/D²×(3×c²/D²+0.25×c²/D⁴) ~ c³/D⁴
Probability of velocity starting at the green cap (comparing to the whole hemisphere of radius c as 100%, which volume is 2/3×π×c³) is proportional to (c³/D⁴) / (2/3×π×c³) ~ 1/D⁴.
As we have mentioned above, combined with decline in frequency of contacting the border by 1/D, that reduces probability of decay/escape by D⁵. In other words, half–life increases by D⁵: atom with 10-times-slower atomic clock lives at least 100,000 times longer (if not forever, when parts have no energy or angle enough to escape through a higher and narrower barrier)!
