Measuring Time Dilation from 25Kg Lead Brick w/3 Cesium Oscillators

[Dc2LightTech]

I have access to 3 Cesium 10MHz Oscillators. If I place one on top, one under and one 10' from a 25Kg lead brick what would to the ns/time rate of change? I am thinking this would be a grate HS Science Fair project.

 

[anuttarasammyak]

Are you considering GR effect by mass of the lead ? If so I think that of the Earth surpass it.

 

[Ibix]

Approximating the mass as a sphere, the ratio of tick rates between clocks at distances $r_1$ and $r_2$ from the center is approximately $1+\frac{GM}{c^2}\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$, where $M$ is the mass of the block and $G$ and $c$ are Newton's gravitational constant and the speed of light, respectively. In SI units, $G/c^2\approx 7\times 10^{-28}\mathrm{m\ kg^{-1}}$. If memory serves, good atomic clocks are stable at about one part in $10^{14}$, so you need your $M/r$ to be of the order $10^{14}$ to bring the tick rate variation up to detectable values. With your $r$ values being of order 1, that means you need about a hundred billion metric tons of lead. Not approximating the mass as a sphere would change the numbers a little, but would make no material difference to the answer

It's not practical. Sorry.

Do you know how precise your particular clocks are? And how transportable are they? Will they work in the cargo space of a moving truck? Something like the Hafele-Keating experiment might be more achievable.

 

[Dc2LightTech]

thanks for the reply, what if I run if for 3 months (or 3 million seconds) and check for total drift? I need to check the long term drift specs. so what would be the time differential from to to bottom of a 22Kg lead brick after 3 months. I can work with > 10PPM or 10us.
This year I am working with a HS Physics teacher that said I may have 12 AP Calc students to work with. I want to bring them into my Engineering lab and to projects that would be beyond the reach of most HS Science projects. Last year I had two students doing gradient fields in copper and took 2nd place.

 

[Vanadium 50]

It's not like you are off by a factor of 1.5 or 2. You're off by a factor of trillions.

CSAC's are good, but not impossibly good. First, a 10 MHz clock "ticks" at 100 ns. Seeing effects smaller than this requires a more complicated setup, which produces their own errors. But you don't have the sensitivity to run Haefle-Keating on a plane, much less a truck (where the effect is 100x smaller)

Academic CSACs are good to 4 x 10^-11/√t (and commercial ones are probably less good). The sorts of time dilation you get on trucks is 10^-15 so it will take decades before you see any effect. And, as pointed out, this effect is orders of magnitude larger than what you propose.

 

[Ibix]

what if I run if for 3 months (or 3 million seconds) and check for total drift?

Accumulated time difference over that period is about $10^{-19}\mathrm{s}$.

 

[Vanadium 50]

Accumulated time difference over that period is about 10^-19s.

a 10 MHz clock "ticks" at 100 ns.

There's that factor of a trillion again.

 

[Dc2LightTech]

so there is no way to induce a relative drift from one to the other of more that 100ns over 3 months? I was thinking of using decade counters and latching the difference, and measuring the phase of the the 2 outputs.

https://www.microchip.com/en-us/product/CSAC-SA65
±5.0E-11 accuracy at shipment
1.5 E-11 @1000s Short Term Stability (Allan Deviation)
9E-10 /mo Aging Rate (Typical)
10MHz square wave and 1PPS, both in a CMOS 0V to 3.3V format.
1 PPS input for synchronization

 

[Vanadium 50]

This is not going to work, but there are still lessons to be learned applicable to many experiments.
1. Learn to do the calculations yourself. You need to be able to ask yourself "what would happen if I changed this parameter to that parameter?" and find the answer quickly.

2. Understand scaling. Look at @Ibix message. Everything is proportional to M/r. Because M ~ r³, your signal is proportional to r². That tells you several things:

• You want a big setup - as big as you can make it.
• Your effect is small compared to the earth, so you want to work horizontally and not vertically.
• Getting in close is advantageous.

3. You need to think about what else can go wrong. For example, cable propagation times are a few 10^-5 or 10^-6 per degree. If you aren't careful, you would build yourself a very expensive thermometer.

If I were serious about this, I'd be thinking aboutr water tanks as my souece, because they are easy ti find large ones, water is cheap, you can place the clock inside the tank (adequately protected, of course) and you can run control tests without water. Howeever, this is still unlikely to work with the precision of timekeeping you have available. Calculation is left as an exercise for the student.

 

[Vanadium 50]

so there is no way to induce a relative drift

@Ibix says the effect is one part in 10¹⁹. From the spec sheet, your clock is good to no better one part in 10¹².

You tell us.

 

[PeterDonis]

so there is no way to induce a relative drift from one to the other of more that 100ns over 3 months?

Not with the specifications you gave.

 

[PeterDonis]

With your values being of order 1, that means you need about a hundred billion metric tons of lead.

Even that won't work because you would need to squeeze all that lead into a 1 meter radius sphere, and lead's density is about 10 orders of magnitude too small (about 10 tons per cubic meter, where you need $10^{11}$). $r$ is the distance from the center of the gravitating mass, not from its surface.

 

[PeterDonis]

Everything is proportional to M/r. Because M ~ r3, your signal is proportional to r2.

Only if you scale up $M$ accordingly as $r$ increases. And given the accuracy of the OP's clocks, you would need to scale things up to roughly the size of a small planet.

 

[Vanadium 50]

size of a small planet.

Well, maybe a good sized moon.

But scaling tells you that as well. It tells you if you need to be three times better or three trillion times better.

Scaling also tells you that lead is only 10x better than water - and it's easier and cheaper to get 10 tons of water than 1 ton of lead.

It's never going to work with these clocks, which frankly, aren't that good. You can get OXCOs that are about this good (and for a lot less money).

With water tanks you can put the clocks (which need to be more sensotive than what the OP has on hand) inside. This buys you what looks like a factor of 3/2 over extrernal clocks, but is more like a factor of 2 since the clocks are not at infinity. I imagine having several tracks with several (>=4) clocks so they can be run in different positions. Looking for the right position dependence will allow me to tune out any residual effects from the horizontal ports not being exactly horizontal. That's going to be the trick - I can tell a survey target's location well enough, but the actual senor element may "wiggle'.

In short, I think a measurement is doable, but not by amateurs with whatever they have laying around. It will not be easy, and it will not be cheap, and - this is the reason why it won't happen - it's not interesting. It would not tell us anything we don't already know. If we could do this with ppm accuracy on the effect (which we can't) it becomes interesting, as a measure of G.

 

[ersmith]

If the OP wants to measure gravitational time dilation, then rather than trying to induce it with a ton of lead (which is impractical) perhaps they should refocus and look at the difference in the passage of time at different altitudes. If they have a mountain nearby they may be able to reproduce the experiment described at http://leapsecond.com/great2005/, where an amateur took some atomic clocks up a mountain for a weekend and observed that they ticked faster.

 

[Vanadium 50]

The problem with that is that CSACs aren't good enough. The effect reported in Physics Today is about 10^-13 and the clocks are good to a few 10^-11.

 

[ersmith]

The experiment I linked to found a 23ns difference between clocks after 2 days. If the OP is willing to run for 3 months, and has access to a similar gravitational potential difference (5400 meters) then the difference should be on the order of 900ns, well above the 100ns threshold they asked for. Mind you, it would take a great deal of care to tease out the effect from background noise -- it's by no means an easy experiment. But it's at least in the ballpark of being feasible, unlike the original proposal involving lead bricks

 

[Vanadium 50]

Unfortunately, if you want to do twice as well, you need to run 4x as long. so he would need to run for centuries, not days.

I don't have one (but am seriously thinking of ordering a few) so I don't know how long they hold lock. Probably not months. Probably days.

Really, CSACs are just not that good. They have their uses, but this is not one of them.