B How Does Speed Influence Time and Particle Decay in Einstein's Theory?

[PeterDonis]

There are potentially mechanical systems that can be controlled well enough that the initial condition of the die pre-toss will show up statistically in the result.

Yes, but that's not the same as the die itself storing information about the results of previous tosses. The correlation between tosses is due to the throwing system, not the die itself.

 

[calinvass]

Does it? If it does, why is it that the probability of a given face coming up doesn't change from one throw to the next?

I'm not sure if it is this what you ask. The example was with a throw every second, and between seconds no throw, but this is only a mechanism that shows the probability for getting one state doesn't change over time, but the mean life of the die can be calculated as 6 seconds I guess.

 

[calinvass]

Possibly because the human hand cannot repeat the throw well enough to achieve any correlation.

There are potentially mechanical systems that can be controlled well enough that the initial condition of the die pre-toss will show up statistically in the result.

This is perhaps your point, that just looking at the die in isolation instead of the entire process involved in a toss can be misleading.

I'm not thinking of a real world object. In this example the die is a perfect system that gives completely random results although a real world one is also almost perfect I think.

 

[PeterDonis]

the probability for getting one state doesn't change over time

Yes, and my question was how that can be true if the structure of the die "memorizes" the last state, as you claimed it did in post #46. I don't think that claim is correct; a die, assuming it's fair, will not "memorize" any information from previous throws.

the mean life of the die can be calculated as 6 seconds I guess

Not quite. The half life (I use that because it's easier to conceptualize) will be the number of throws after which there is a 50% probability of throwing a 1 (or whichever face we treat as "destroying" the die when it comes up). The easiest way to calculate that is to find the smallest exponent $N$ such that

$$
\left( \frac{5}{6} \right)^N < \frac{1}{2}
$$

In other words, how many throws does it take before there is a less than 50% chance of not throwing a 1? (This method is easier because there is only one term on the LHS of the equation.) I get $N = 4$ as the smallest integer exponent (the corresponding probability, of not throwing a 1 in 4 throws, is about 48 percent). So the "half life" of a die is about 4 throws (a little less if we want an exact floating point answer).

 

[Dale]

I don't think that the fact that it is a "memoryless" Poisson process can be used alone to imply that there is no internal structure

I agree. The implication is the other way around. If it has no internal structure then its chance of spontaneously decaying must be memoryless. We postulate that muons have no internal structure, which implies a memoryless decay probability, we observe such a decay probability, which we take to be evidence supporting the postulate.

 

[calinvass]

Yes, and my question was how that can be true if the structure of the die "memorizes" the last state, as you claimed it did in post #46. I don't think that claim is correct; a die, assuming it's fair, will not "memorize" any information from previous throws.

Yes, you are right, but still its state changes every time. It doesn't make any progress over time but its state changes and that can be thought as a change in its structure I suppose.

 

[Dale]

its state changes and that can be thought as a change in its structure I suppose

Not its internal structure.

There are two memoryless distributions, one for continuous and one for discrete. Muons decay according to the continuous one, dice roll according to the discrete one.

 

[Grinkle]

I am finding the die analogy confusing. Each toss is self-contained and there is no toss-to-toss memory. Within a single toss there is obvious structure. The die tumbles, different faces are 'up' during the toss process, and finally the die comes to rest. Only some state transitions are possible, the die cannot flip from 6-dots to 1-dot without having moved through some other intermediate state, for instance.

What is the comparison to muon decay? Every half life is a toss, and we don't know what happens inside the half life interval, but we do know that each half life behaves as though it is the only half life to have ever expired? If that is the comparison, it does not lead me to any inclination that muon's have no internal structure, since a single die toss clearly does.

 

[FactChecker]

Whether something has an internal structure is not the same question as whether it has "memory" of its state. A die is in a state which includes its orientation and velocities. It retains that state till it is changed by some external force. That is "memory".
Suppose a die is sitting with the 5 side up and the next roll ended with the 5 side up. Now suppose the die had started with the 3 side up. With the exact same external influences, the roll would end with the 3 side up. That means that the die itself has a state and "memory" of it that influences the next result. The die having no internal structure does not prevent that. The randomness of rolling a die is due to external forces that are treated as random changes of its state . We just don't (or can't) keep track of the "random" external influences.

 

[PeterDonis]

but still its state changes every time

In the sense that which face is up changes, yes. (Or, if you want to keep track in more detail, the die's orientation and velocity changes with time.) But, as Dale and FactChecker have pointed out, this is not the same as the die's internal structure changing or the die having "memory" of previous throws stored in its internal structure.

 

[PAllen]

here is the referenced link from a wiki article that said laboratory tests have it to less than a meter.

Actually, the one meter tests are NOT detecting tidal gravity. They are detecting potential difference over one meter, and cannot distinguish the Earth surface observation from a uniformly accelerating rocket in empty space - the results of the experiment would be identical. Note, that Peter's computation shows you would need several significant digits in precision of time dilation measurement over a kilometer to distinguish tidal gravity. These experiments are at detection limit at one meter, e.g one significant digit. Instead they would need e.g 10 to detect tidal gravity.

[edit: well, a series of measurements from such clocks can verify tidal gravity. Separate such clocks radially versus tangentially, and do this on opposite sides of the earth, to take the extreme. Then only tidal gravity can explain the collection of observations. But the, again, we are expanding the scope of observations greatly. The key point being that if observations are sufficiently local, you cannot distinguish.]

 

[nitsuj]

Actually, the one meter tests are NOT detecting tidal gravity. They are detecting potential difference over one meter, and cannot distinguish the Earth surface observation from a uniformly accelerating rocket in empty space - the results of the experiment would be identical. Note, that Peter's computation shows you would need several significant digits in precision of time dilation measurement over a kilometer to distinguish tidal gravity. These experiments are at detection limit at one meter, e.g one significant digit. Instead they would need e.g 10 to detect tidal gravity.

[edit: well, a series of measurements from such clocks can verify tidal gravity. Separate such clocks radially versus tangentially, and do this on opposite sides of the earth, to take the extreme. Then only tidal gravity can explain the collection of observations. But the, again, we are expanding the scope of observations greatly. The key point being that if observations are sufficiently local, you cannot distinguish.]

I don't know enough about that distinction, but was replying to Peter's comment "Even if 1 km is too large, there will be some height change which is small enough that no effects of spacetime curvature are measurable, though."

I do know the distinction between flat spacetime and curved spacetime, and this experiment shows that we have this measurable to less than a meter. What of tidal gravity or difference in gravitational potential...when neither are flat.

 

[PAllen]

I don't know enough about that distinction, but was replying to Peter's comment "Even if 1 km is too large, there will be some height change which is small enough that no effects of spacetime curvature are measurable, though."

I do know the distinction between flat spacetime and curved spacetime, and this experiment shows that we have this measurable to less than a meter. What of tidal gravity or difference in gravitational potential...when neither are flat.

No, this measurement does not distinguish curvature at all over one meter. It would give the exact same result in flat spacetime in an accelerating rocket.

 

[nitsuj]

I am finding the die analogy confusing. Each toss is self-contained and there is no toss-to-toss memory. Within a single toss there is obvious structure. The die tumbles, different faces are 'up' during the toss process, and finally the die comes to rest. Only some state transitions are possible, the die cannot flip from 6-dots to 1-dot without having moved through some other intermediate state, for instance.

I'm finding the distinction between "memory" and causal connection confusing.

 

[nitsuj]

No, this measurement does not distinguish curvature at all over one meter. It would give the exact same result in flat spacetime in an accelerating rocket.

Sorry PAllen, I still fail to see the issue. You're mentioning the equivalence principal. So in a sealed free fall laboratory, this experiment doesn't work...neat physics.

 

[PAllen]

Sorry PAllen, I still fail to see the issue. You're mentioning the equivalence principal. So in a sealed free fall laboratory, this experiment doesn't work...neat physics.

I will try again to clarify. Uniform acceleration in flat spacetime is distinguishable from behavior near Earth over sufficient distances or time. Peter noted that for one kilometer, it should be possible to make the distinction. You claimed this experiment shows a distinction over one meter. This is simply false. This experiment could not distinguish the time dilation over one meter that would occur in an accelerating rocket from that on the Earth's surface. To make that distinction would require many order of magnitude greater precision. In, particular, a device that could measure presence of time dilation over e.g. one millimeter might stand a chance of detecting curvature over one meter.

 

[PeterDonis]

You're mentioning the equivalence principal. So in a sealed free fall laboratory, this experiment doesn't work

The correct comparison is not with a sealed free fall laboratory, but with a sealed rocket accelerating at 1 g in flat spacetime. And PAllen is saying, correctly, that the results of such an experiment would be indistinguishable from those of the experiment as it was performed on the Earth's surface. In other words, the experiment shows time dilation but does not show tidal gravity--it isn't precise enough for the latter.

 

[Herman Trivilino]

Yes, you are right, but still its state changes every time. It doesn't make any progress over time but its state changes and that can be thought as a change in its structure I suppose.

In the same way, the decay of a muon is a change in its state.

I am finding the die analogy confusing.

Each roll of the die is analogous to an observation of the muon. When you roll the die it either will or will not show "1". When you observe a muon it either will or will not decay.

You can find break-downs in the analogy. That's easy. Because it's an analogy.

Let's remember that the reason for the analogy was to try to understand why a particle such as a muon can decay despite the fact that it has no internal structure. The point was made that unlike humans, the decay rate stays the same over time, which is something we notice only after looking at large numbers of both humans and muons decaying. But that still doesn't explain how something with no internal structure can decay. It may be that there is no explanation, in which case we'd have to ask ourselves why we'd expect one to exist!

 

[nitsuj]

The correct comparison is not with a sealed free fall laboratory, but with a sealed rocket accelerating at 1 g in flat spacetime. And PAllen is saying, correctly, that the results of such an experiment would be indistinguishable from those of the experiment as it was performed on the Earth's surface. In other words, the experiment shows time dilation but does not show tidal gravity--it isn't precise enough for the latter.

Thanks PAllen and Peter, I see the accelerating rocket compared to Earth equivalence. And if I see that I got to say it's not measuring tidal gravity. The experiment simply determines the differential in gravitational potential, which is (this is where all my fog is) distinctly different from tidal gravity.I not good at math, or even keeping track of significant digits. but as far as the measurement accuracy in this experiment even if there were four measures taken, say one meter square area, The horizontal (to earth) comparatives wouldn't deviate, but the vertical ones do? Compared to "flat space", i intuit a trapezoid shape.

 

[PeterDonis]

The experiment simply determines the differential in gravitational potential, which is (this is where all my fog is) distinctly different from tidal gravity.

Yes, this is correct.

The horizontal (to earth) comparatives wouldn't deviate, but the vertical ones do?

Tidal gravity on Earth shows up in both the horizontal and vertical directions. But the horizontal change is in the direction of gravity, whereas the vertical change is in its magnitude. So you might use different methods, with different accuracies, to measure the two.

 

[Herman Trivilino]

Sorry PAllen, I still fail to see the issue. You're mentioning the equivalence principal. So in a sealed free fall laboratory, this experiment doesn't work...neat physics.

Detecting a potential is not the same as detecting a potential gradient. If the time dilation effect you referenced is different when detected at sea level than at an altitude of 1.0 km, that would demonstrate tidal gravity. Just detecting the effect at sea level, or at the 1.0 km altitude, isn't enough.

In other words stand a meter stick up on end at sea level, and do the same with another at the top of a mountain. You can detect the time dilation effect between the ends of the sea-level meter stick, and the mountain-top meter stick. And perhaps note that it's not the same for each meter stick. Thus the time dilation effect is different at sea level than it is on the mountain top. That's a determination of tidal gravity.

 

[nitsuj]

Detecting a potential is not the same as detecting a potential gradient. If the time dilation effect you referenced is different when detected at sea level than at an altitude of 1.0 km, that would demonstrate tidal gravity. Just detecting the effect at sea level, or at the 1.0 km altitude, isn't enough.

In other words stand a meter stick up on end at sea level, and do the same with another at the top of a mountain. You can detect the time dilation effect between the ends of the sea-level meter stick, and the mountain-top meter stick. And perhaps note that it's not the same for each meter stick. Thus the time dilation effect is different at sea level than it is on the mountain top. That's a determination of tidal gravity.

lol i still don't get it...not at all an issue with your explanation...that reads easily.

 

[PAllen]

Detecting a potential is not the same as detecting a potential gradient. If the time dilation effect you referenced is different when detected at sea level than at an altitude of 1.0 km, that would demonstrate tidal gravity. Just detecting the effect at sea level, or at the 1.0 km altitude, isn't enough.

In other words stand a meter stick up on end at sea level, and do the same with another at the top of a mountain. You can detect the time dilation effect between the ends of the sea-level meter stick, and the mountain-top meter stick. And perhaps note that it's not the same for each meter stick. Thus the time dilation effect is different at sea level than it is on the mountain top. That's a determination of tidal gravity.

And to get really technical, time dilation per meter would be expected to change with position within a long uniformly accelerating rocket, per the Rindler metric, which is still flat spacetime. To truly distinguish this from Earth gravity you would need to measure the deviation in this change compared to Rindler. This is a second order effect.

 

[Dale]

Yes, you are right, but still its state changes every time. It doesn't make any progress over time but its state changes and that can be thought as a change in its structure I suppose.

Fundamental particles (muons, electrons, quarks, etc.) can certainly occupy different states. But regardless of what state they are in they have no internal structure. Internal structure and state are very different things.

In that sense the dice is a good analogy. It can be in different states, and those states may be stable, but it doesn't change its internal structure and has no memory.

 

[nitsuj]

And to get really technical, time dilation per meter would be expected to change with position within a long uniformly accelerating rocket, per the Rindler metric, which is still flat spacetime. To truly distinguish this from Earth gravity you would need to measure the deviation in this change compared to Rindler. This is a second order effect.

Wiki says the tensile force from tidal gravity for a 1 meter, 1kg rod on Earths surface is about 0.4uN...and apparently that's measurable. I wonder if ideally an experiment to measure tidal gravity could be done with these "things". I wonder if we can make a string that breaks at a specific tensile force value, where the only explanation for it breaking is tidal forces.

Same article says the tensile force on the same rod on the surface of a white dwarf (mass of sun, size of earth) is 0.24N lol tidal gravity is so subtle...they gave an extreme example of a neutron star (2 sun mass, 12km radius) where the tensile force on the rod is 10,000N 190km about the surface, that's some strong geometry!

 

[PAllen]

Wiki says the tensile force from tidal gravity for a 1 meter, 1kg rod on Earths surface is about 0.4uN...and apparently that's measurable. I wonder if ideally an experiment to measure tidal gravity could be done with these "things". I wonder if we can make a string that breaks at a specific tensile force value, where the only explanation for it breaking is tidal forces.

Same article says the tensile force on the same rod on the surface of a white dwarf (mass of sun, size of earth) is 0.24N lol tidal gravity is so subtle...they gave an extreme example of a neutron star (2 sun mass, 12km radius) where the tensile force on the rod is 10,000N 190km about the surface, that's some strong geometry!

That is valid detection of tidal gravity. I suspect the smallest scale detection would be achieved with SQUID based accelerometers. Time dilation is just not a very practical way to detect tidal gravity.

 

[Herman Trivilino]

And to get really technical, time dilation per meter would be expected to change with position within a long uniformly accelerating rocket, per the Rindler metric, which is still flat spacetime. To truly distinguish this from Earth gravity you would need to measure the deviation in this change compared to Rindler. This is a second order effect.

So in GR you can have a potential gradient that is not due to tidal gravity? This is not so in the Newtonian approximation, correct?

 

[jbriggs444]

So in GR you can have a potential gradient that is not due to tidal gravity? This is not so in the Newtonian approximation, correct?

Is it just me? The term "potential gradient" sounds like a NOOP based on the fundamental theorem of calculus.

 

[Ibix]

So in GR you can have a potential gradient that is not due to tidal gravity? This is not so in the Newtonian approximation, correct?

I think @PAllen's point is that if you built a 1km long rocket and accelerated it at 1g, then got two clocks attached to opposite ends of a 1m rule then you would always find the same relative tick rate between the clocks as you climbed deck by deck. However, if you built a 1km tall tower on Earth and climbed it with the same pair of clocks you would notice different relations between the tick rates as you climbed.

With today's clock technology we could not repeat the experiment with a 1m rocket/tower and two clocks 1mm apart, although in principle the same type of results would be expected. This is why he's saying that the 1m measurement is of a potential gradient (present in Rindler coordinates in the rocket and in the tower) and not of tidal gravity (present in the tower and not the rocket).

 

[PAllen]

So in GR you can have a potential gradient that is not due to tidal gravity? This is not so in the Newtonian approximation, correct?

Yes, a born rigid rocket will have lower proper acceleration at the front compared to the back, under constant thrust = constant proper acceleration of the back. In equlibrium, a real rocket would be expected to exceedingly well approximated by born rigidity.

 

[PAllen]

I think @PAllen's point is that if you built a 1km long rocket and accelerated it at 1g, then got two clocks attached to opposite ends of a 1m rule then you would always find the same relative tick rate between the clocks as you climbed deck by deck. However, if you built a 1km tall tower on Earth and climbed it with the same pair of clocks you would notice different relations between the tick rates as you climbed.

With today's clock technology we could not repeat the experiment with a 1m rocket/tower and two clocks 1mm apart, although in principle the same type of results would be expected. This is why he's saying that the 1m measurement is of a potential gradient (present in Rindler coordinates in the rocket and in the tower) and not of tidal gravity (present in the tower and not the rocket).

No, that's not true. See my reply above.

[edit: I should say, most of what you say is clearly correct and what I meant. However, born rigidity leads to change in 'g' with position in a uniformly accelerating rocket, but with a different, slower, functional dependence than 'g' as a function of altitude near earth.]

 

[Ibix]

No, that's not true. See my reply above.

[edit: I should say, most of what you say is clearly correct and what I meant. However, born rigidity leads to change in 'g' with position in a uniformly accelerating rocket, but with a different, slower, functional dependence than 'g' as a function of altitude near earth.]

Ah - because (in an inertial frame) the rocket is doing something like length contracting more and more, so the acceleration is non-uniform even in the steady state. Right?

 

[PAllen]

Ah - because (in an inertial frame) the rocket is doing something like length contracting more and more, so the acceleration is non-uniform even in the steady state. Right?

Right.

 

[Dale]

So in GR you can have a potential gradient that is not due to tidal gravity? This is not so in the Newtonian approximation, correct?

In Newtonian gravity the usual mgh formula is the potential gradient in a uniform field (no tidal effects).

Edit: oops, mgh is the potential energy, the potential is gh, and the potential gradient is g. See below

 

[Herman Trivilino]

In Newtonian gravity the usual mgh formula is the potential gradient in a uniform field (no tidal effects).

Right. I knew I had gotten something wrong in my thinking when I read @PAllen 's response in which he talked about the potential gradient in a rocket. So the issue is whether or not the potential gradient is uniform. One would need to measure how the potential gradient changes as one ascends that mountain. If it changes uniformly there's no tidal gravity, but if the decrease is nonlinear then that would be a detection of tidal gravity.

 

[jartsa]

Right. I knew I had gotten something wrong in my thinking when I read @PAllen 's response in which he talked about the potential gradient in a rocket. So the issue is whether or not the potential gradient is uniform. One would need to measure how the potential gradient changes as one ascends that mountain. If it changes uniformly there's no tidal gravity, but if the decrease is nonlinear then that would be a detection of tidal gravity.

So we measure the g, right?
So, potential gradient = g
The g does not change uniformly in an accelerating rocket. How fast the g changes is
proportional to g squared. In an accelerating rocket and in an uniform gravity field the g is not uniform, and the change of g is not uniform either.

(g = gravitational acceleration)

 

[David Lewis]

So, potential gradient = g

It might be gh. (It takes energy to move in the direction of motion in an accelerating rocket.)

 

[jbriggs444]

It might be gh. (It takes energy to move in the direction of motion in an accelerating rocket.)

Gravitational potential is approximately gh near the surface of the Earth.

Gravitational force is the gradient of the potential and is, in the Newtonian approximation, $\frac{GM}{r^2}$ which is equal to g at the Earth's surface. Tidal gravity is about how the gravitational force changes with position -- i.e. whether the gravitational force field has a non-vanishing curl.

The term "potential gradient" in this context is a synonym for the gravitational force field -- the gradient of the gravitational potential. So the "potential gradient" is indeed equal to g, not gh.

 

[Dale]

Gravitational potential is approximately gh near the surface of the Earth.

Gravitational force is the gradient of the potential and is, in the Newtonian approximation, $\frac{GM}{r^2}$ which is equal to g at the Earth's surface. Tidal gravity is about how the gravitational force changes with position -- i.e. whether the gravitational force field has a non-vanishing curl.

The term "potential gradient" in this context is a synonym for the gravitational force field -- the gradient of the gravitational potential. So the "potential gradient" is indeed equal to g, not gh.

Oh, you are right. I have added an edit to my incorrect post above. Sorry for any confusion I caused!

 

[vanhees71]

Gravitational potential is approximately gh near the surface of the Earth.

Gravitational force is the gradient of the potential and is, in the Newtonian approximation, $\frac{GM}{r^2}$ which is equal to g at the Earth's surface. Tidal gravity is about how the gravitational force changes with position -- i.e. whether the gravitational force field has a non-vanishing curl.

The term "potential gradient" in this context is a synonym for the gravitational force field -- the gradient of the gravitational potential. So the "potential gradient" is indeed equal to g, not gh.

In Newtonian theory the gravitational field is a potential field, i.e., its curl is vanishing. The field equation of motion reads
$$\vec{\nabla} \cdot \vec{G}=-4 \pi \gamma \rho, \quad \vec{G}=-\vec{\nabla} g,$$
where $\vec{F}$ is the gravitational field, $\gamma$ Newton's gravitational constant, $\rho$ the mass density of the gravitating body and $g$ the potential of the gravitational field. The first equation says that the mass density of matter is the source of the gravitational field, and the 2nd equation says that it's a curl-free field. Finally you get
$$\Delta g=4 \pi \gamma \rho.$$
The solution is given by the Green's function of the Laplace operator, leading to
$$g(\vec{x})=-\gamma \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \frac{\rho(\vec{x}')}{|\vec{x}-\vec{x}'|}.$$
For a point source in the origin, $\rho(\vec{x})=M \delta^{(3)}(\vec{x})$ you get, of course,
$$g(\vec{x})=-\frac{\gamma}{r}, \quad \vec{G}=-\vec{\nabla} g=-\gamma \frac{\vec{x}}{r^3}, \quad r=|\vec{x}|.$$
The gravitational force on a test particle of mass $m$ is
$$\vec{F}_{\text{G}}(\vec{x})=m\vec{G}(\vec{x})=-m \vec{\nabla} g(\vec{x}).$$

 

[Herman Trivilino]

Gravitational force is the gradient of the potential and is, in the Newtonian approximation, $\frac{GM}{r^2}$ which is equal to g at the Earth's surface.

That's the gravitational field, the force is $mg$.

 

[jbriggs444]

That's the gravitational field, the force is $mg$.

Yes. I tend to speak a little fast and loose because there is no good pithy term for the force-per-unit-mass.