Theory – Curved Time Theory

26 December 2016

Introduction

A lot of models of the Universe and interpretation of experimental data is made with the fundamental (and often unstated) assumption that a clock, or time, that remains in a single frame of reference, will tick along at constant rate (in reference to itself, or another clock also remaining in a constant frame of reference). What if this is not the case, and a clock remaining in a constant frame of reference accelerates at a slow, but exponential rate? It might mean quite a lot, and could change the way we think about cosmology, relativity, and even gravity. Read on if you find this interesting. Below, I flush out the idea very briefly, and then propose a simple experiment to test it.

Pushing Marbles

What do I mean when I propose that time could be accelerating? Allow me to explain. It was proven by Einstein that clocks can tick along at different rates. Two causes of this are fast travel and gravity wells. A clock traveling through the cosmos at high speeds will tick slower than a clock that is not. A clock that’s in a gravity well will also tick slower. What is a gravity well? It’s any big object that has a lot of gravity and therefore bends space-time, like the Earth for instance. So if you and your brother are sharing bunk beds, you could speed up your clock by always taking the top bunk. This would give you an edge in growing up faster, and also give you more rest time. Of course this edge would be so small as to be nearly immeasurable, but this phenomenon is very real and has to be taken into account in order for GPS satellite technology to work correctly.

Back to the bunk beds… So if, from the top bunk, you were pushing marbles to your brother, (who is on the bottom bunk) at a constant rate, he would receive those marbles at a slightly faster rate than you sent them (sort of—really his clock is just slower). Conversely, if he were pushing marbles to you, you would get them at a slower rate than he was sending them. The assumption is, though, that if you were to send him marbles at a specific rate, and he were to turn around and give each marble back to you when he got it, that you would get the marbles back at the same rate you sent them. What if this was not the case and the two time dilations didn’t cancel each other out perfectly? Well, if you got the marbles back faster than you sent them, you could just make marbles out of thin air. So let’s rule that idea out because I think it would break the Universe. But what if you got the marbles back at a slightly slower rate? Does this break anything? Let’s try it. From the top bunk, we will send out marbles at a rate of 10 marbles per minute. They each travel along the same fixed path and eventually come back to us, but at a slightly slower rate—say 9 marbles per minute. When I was explaining this to my wife, she observed correctly that I was losing my marbles. If we keep doing this for some time, assuming we don’t have an infinite supply of marbles, we will eventually run out of marbles. We will then be in the position of waiting for a marble to come back in order to continue the experiment. If we decide to continue, we will have to slow the marble push rate to 9 marbles per minute. Or we could stop the experiment entirely and eventually we will get all of our marbles back. Strange results, but the Universe remains unbroken at the end of the experiment.

So this, then, is what I mean when I say that time is accelerating. I mean that if, while remaining in a constant frame of reference, I send out marbles at constant rate, and each marble undergoes an identical round-trip journey, that I will receive the marbles back at a slower rate than the rate at which I sent them.

The Hubble Constant

How fast is time accelerating? If we use the redshifted light coming from distant galaxies as a measure, the rate of time is increasing by a factor of $7 \times 10^{-11}$ per year. How did I arrive at this number? Let’s talk about Hubble’s Law [1]. From Wikipedia:

“Objects observed in deep space (extragalactic space, 10 megaparsecs (Mpc) or more) are found to have a Doppler shift interpretable as relative velocity away from Earth.
This Doppler-shift-measured velocity, of various galaxies receding from the Earth, is approximately proportional to their distance from the Earth for galaxies up to a few hundred megaparsecs away.”

In the above definition from Wikipedia, the “Doppler shift” is assumed because things that occur at certain frequencies of light are showing up at lower frequencies when we see them from very far away. Hence those objects must be moving away from us. But what if the frequency shift is not occurring because of a Doppler Shift, but because the light is coming from long ago when time was slower? The computed rate of expansion, or “Hubble Constant” varies per experiment, but it is generally about $70km/s /Mpc$ , or seventy kilometers per second per megaparsec [1]. Using the “Fieau-Doppler Formula” as an approximation[1]:

$z=\frac{\lambda-\lambda_0}{\lambda_0}=\frac{v}{c}$

where v is the velocity the object is traveling away from us, c is the speed of light, $\lambda_0$ is the proper wavelength for a known thing (like hydrogen spectral lines), and $\lambda$ is the wavelength measured. If we substitute frequency in for lamba ( $f=c/\lambda$ ) and do some math, we get:

$\frac{\Delta f}{f} \approx \frac{v}{c}=\frac{70km/s}{3 \times 10^8m/s}=2.\overline{3} \times 10^{-4}$

So we’re looking at a $2.\overline{3} \times 10^{-4}$ factor decrease in frequency per megaparsec that is being interpreted as a Doppler shift. Let’s choose to interpret the frequency shift in a different way. One megaparsec is approximately equal to 3.3 million light years. If we assume a constant rate of increase (compounding like interest) then:

$R=e^{\frac{ln(1.0002\overline{3})}{3.3 \times 10^6}}-1=7.07 \times 10^{-11}$

We arrive at a $7.07 \times 10^{-11}$ factor increase in clock rate per year. Note that, at values of z << 1, we can approximate the equation above by treating the rate of increase as linear. Case in point: $2.\overline{3} \times 10^{-4}/3.3 \times 10^{6} = 7.07 \times 10^{-11}$ . We can do this because at values very close to each other, the exponential curve can be treated as flat. Using this simplification:

$R=e^{\frac{ln(1+7.07\times 10^{-11} )}{365 \times 24 \times 60 \times 60}}-1\approx \frac{7.07 \times 10^{-11}}{365 \times 24 \times 60 \times 60}=2.24 \times 10^{-18}$

The Experiment

I propose we conduct an experiment. Not a mind experiment like the marbles, but a real experiment with measurable results. The result will either support the Hubble theory of an expanding universe, or it will support the idea that the rate of time is accelerating.

Let’s start with the fastest known atomic clock on the planet. According to the Guinness Book of World Records, the fastest atomic clock runs at a rate of 1 petaHz (that’s $10^{15}$ Hz) or $3.1563 \times 10^{22}$ cycles per year, or Yerts (Yz) as I like to call them. We will attempt to measure an increase in frequency of 1 Yz. So with our fast clock, using the R-per-second rate of time acceleration from above, after one second, our super fast clock will see an acceleration of:

$3.1536 \times 10^{22} \times R = 3.1536 \times 10^{22} \times 2.24 \times 10^{-18} = 70,641 Yz$

So if we want to measure an increase in speed of 1 Yz, we will need to delay the clock by 1/70641 seconds. 1/70641 light seconds is equal to 4.25 km.

To conduct the experiment we will use a high frequency laser to carry the frequency of the clock. We split the laser into two paths. One path takes a 4.25 km round trip through a fiber optic cable. We combine the two beams again—one having just exited the clock, the other one being 1/70641 seconds old. If time is accelerating at the rate computed, we should see an interferometric pulse beat in about one year. If instead, Hubble’s Law is correct and the Universe is expanding, we should expect to see no change in clock frequency.

Conclusion

What would the results mean? If the result showed no shift in clock rate over time, then clock-rate acceleration can be ruled out as a variable. If the results are similar to the prediction based on previously observed redshift of far away galaxies, then there would be some fun work to do. The actual Doppler shift and the redshift from clock acceleration should sum together. So if the clock acceleration is accounted for and subtracted from the total shift, we should be left with the true Doppler shift. Is the Universe still expanding, but slower than we thought? Or possibly we would see that it’s actually contracting. My guess is that, once the adjustment is made we will see a little bit of both redshift and blueshift—similar to what we see in “local” galaxies. This could be the reason for the apparent different values measured for the Hubble constant from different experiments.

Also, it would be important to do this same test in other frames of reference. For instance, the same experiment should be conducted under low gravity conditions.

One more thing… This would also mean the Universe no longer has time-symmetry. So, probably no more backwards time travel. Sorry.

References

[1] https://en.wikipedia.org/wiki/Hubble%27s_law