# Time Dilation and Particle Decay

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I am teaching myself astronomy and I'm working on a basic problem regarding time dilation around black holes.

The question is basically this: if a neutron were ejected from a nucleus at a distance of 3km from a 1 solar mass black hole, how long would it appear to take for the neutron to decay (the assumption here is that the neutron would decay in 15 minutes… right on the half-life) to an outside observer?

My first thought was that we'd never see it decay, because 3km is the Schwarzschild radius of a BH of that mass, so the frequency would basically be $$0$$ and we'd never observe the decay event.

The text notes the time dilation

$$frac{Delta t'}{Delta t_ ext{obs}} = frac{ u_f}{ u_i}$$ where $$Delta t'$$ is the the time difference between two events at the source and $$Delta t_{obs}$$ at the observer.

We can use the relativistic red shift equation to get the wavelength difference (9.2 in my case using 1 solar mass and a radius of 3km) so that wouldn't be infinite, just big. For a 15 min $$Delta t'$$ then, the $$Delta t_{obs}$$ would be 15 min * 9.2, yeah?

So my two answers obviously contradict each other. What am I missing in my approach here?

If you use the equation of the Schwarzschild metric. $$ds^{2}=-(1-frac{2M}{r})dt^{2}+(1-frac{2M}{r})^{-1}dr^{2}+r^{2}dOmega^{2}$$ (equation 11.1 from Schutz's book A first course in GR).

If we think the neutron stay at rest in $$r=r_{S}=2M$$ then between the born and dead of the neutron $$dr=0$$ and $$dOmega=0$$, and $$ds$$ will be the proper time, which I will call $$au$$ (I'm folowing the convention $$c=1=G$$).

Putting this on the metric we have

$$d au^{2}=-(1-frac{2M}{r})dt_{obs}^{2}$$

$$frac{d au^{2}}{dt_{obs}^{2}}=-(1-frac{2M}{r})$$

and if we put $$r=r_{s}$$ (where the neutros was), we have

$$frac{d au^{2}}{dt_{obs}^{2}}=0$$

Then $$Delta t_{obs}=infty$$ .

I couldn't write a comment that's why I answered, but I think you had some mistake using the red shift equation to get the wavelength difference.

Can you show your calculus? May be I can help.

## Time dilation

In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them (special relativistic "kinetic" time dilation) or to a difference in gravitational potential between their locations (general relativistic gravitational time dilation). When unspecified, "time dilation" usually refers to the effect due to velocity.

After compensating for varying signal delays due to the changing distance between an observer and a moving clock (i.e. Doppler effect), the observer will measure the moving clock as ticking slower than a clock that is at rest in the observer's own reference frame. In addition, a clock that is close to a massive body (and which therefore is at lower gravitational potential) will record less elapsed time than a clock situated further from the said massive body (and which is at a higher gravitational potential).

These predictions of the theory of relativity have been repeatedly confirmed by experiment, and they are of practical concern, for instance in the operation of satellite navigation systems such as GPS and Galileo. [1] Time dilation has also been the subject of science fiction works.

## How To Prove Einstein's Relativity In The Palm Of Your Hand

Cosmic rays, which are ultra-high energy particles originating from all over the Universe, strike . [+] protons in the upper atmosphere and produce showers of new particles. The fast-moving charged particles also emit light due to Cherenkov radiation as they move faster than the speed of light in Earth's atmosphere, and produce secondary particles that can be detected here on Earth.

Simon Swordy (U. Chicago), NASA

When you hold out your palm and point it towards the sky, what is it that's interacting with your hand? You might correctly surmise that there are ions, electrons and molecules all colliding with your hand, as the atmosphere is simply unavoidable here on Earth. You might also remember that photons, or particles of light, must be striking you, too.

But there's something more striking your hand that, without relativity, simply wouldn't be possible. Every second, approximately one muon — the unstable, heavy cousin of the electron — passes through your outstretched palm. These muons are made in the upper atmosphere, created by cosmic rays. With a mean lifetime of 2.2 microseconds, you might think the

100+ km journey to your hand would be impossible. Yet relativity makes it so, and the palm of your hand can prove it. Here's how.

While cosmic ray showers are common from high-energy particles, it's mostly the muons which make it . [+] down to Earth's surface, where they are detectable with the right setup.

Alberto Izquierdo courtesy of Francisco Barradas Solas

Individual, subatomic particles are almost always invisible to human eyes, as the wavelengths of light we can see are unaffected by particles passing through our bodies. But if you create a pure vapor made out of 100% alcohol, a charged particle passing through it will leave a trail that can be visually detected by even as primitive an instrument as the human eye.

As a charged particle moves through the alcohol vapor, it ionizes a path of alcohol particles, which act as centers for the condensation of alcohol droplets. The trail that results is both long enough and long-lasting enough that human eyes can see it, and the speed and curvature of the trail (if you apply a magnetic field) can even tell you what type of particle it was.

This principle was first applied in particle physics in the form of a cloud chamber.

A completed cloud chamber can be built in a day out of readily-available materials and for less than . [+] $100. You can use it to prove the validity of Einstein's relativity, if you know what you're doing! Instructables user ExperiencingPhysics Today, a cloud chamber can be built, by anyone with commonly available parts, for a day's worth of labor and less than$100 in parts. (I've published a guide here.) If you put the mantle from a smoke detector inside the cloud chamber, you'll see particles emanate from it in all directions and leave tracks in your cloud chamber.

That's because a smoke detector's mantle contains radioactive elements such as Americium, which decays by emitting α-particles. In physics, α-particles are made up of two protons and two neutrons: they're the same as a helium nucleus. With the low energies of the decay and the high mass of the α-particles, these particles make slow, curved tracks and can even be occasionally seen bouncing off of the cloud chamber's bottom. It's an easy test to see if your cloud chamber is working properly.

For an extra bonus of radioactive tracks, add the mantle of a smoke detector to the bottom of your . [+] cloud chamber, and watch the slow-moving particles emanating outward from it. Some will even bounce off the bottom!

If you build a cloud chamber like this, however, those α-particle tracks aren't the only things you'll see. In fact, even if you leave the chamber completely evacuated (i.e., you don't put a source of any type inside or nearby), you'll still see tracks: they'll be mostly vertical and appear to be perfectly straight.

This is because of cosmic rays: high-energy particles that strike the top of Earth's atmosphere, producing cascading particle showers. Most of the cosmic rays are made up of protons, but move with a wide variety of speeds and energies. The higher-energy particles will collide with particles in the upper atmosphere, producing particles like protons, electrons, and photons, but also unstable, short-lived particles like pions. These particle showers are a hallmark of fixed-target particle physics experiments, and they occur naturally from cosmic rays, too.

Although there are four major types of particles that can be detected in a cloud chamber, the long . [+] and straight tracks are the cosmic ray muons, which can be used to prove that special relativity is correct.

Wikimedia Commons user Cloudylabs

The thing about pions is that they come in three varieties: positively charged, neutral, and negatively charged. When you make a neutral pion, it just decays into two photons on very short (

10 -16 s) timescales. But charged pions live longer (for around 10 -8 s) and when they decay, they primarily decay into muons, which are point particles like electrons but have 206 times the mass.

Muons also are unstable, but they're the longest-lived unstable fundamental particle as far as we know. Owing to their relatively small mass, they live for an astoundingly long 2.2 microseconds, on average. If you were to ask how far a muon could travel once created, you might think to multiply its lifetime (2.2 microseconds) by the speed of light (300,000 km/s), getting an answer of 660 meters. But that leads to a puzzle.

Cosmic ray shower and some of the possible interactions. Note that if a charged pion (left) strikes . [+] a nucleus before it decays, it produces a shower, but if it decays first (right), it produces a muon that will reach the surface.

Konrad Bernlöhr of the Max-Planck-Institute at Heidelberg

I told you earlier that if you hold out the palm of your hand, roughly one muon per second passes through it. But if they can only live for 2.2 microseconds, they're limited by the speed of light, and they're created in the upper atmosphere (around 100 km up), how is it possible for those muons to reach us?

You might start to think of excuses. You might imagine that some of the cosmic rays have enough energy to continue cascading and producing particle showers during their entire journey to the ground, but that's not the story the muons tell when we measure their energies: the lowest ones are still created some 30 km up. You might imagine that the 2.2 microseconds is just an average, and maybe the rare muons that live for 3 or 4 times that long will make it down. But when you do the math, only 1-in-10 50 muons would survive down to Earth in reality, nearly 100% of the created muons arrive.

A light-clock, formed by a photon bouncing between two mirrors, will define time for any observer. . [+] Although the two observers may not agree with one another on how much time is passing, they will agree on the laws of physics and on the constants of the Universe, such as the speed of light. When relativity is applied correctly, their measurements will be found to be equivalent to one another, as the correct relativistic transformation will allow one observer to understand the observations of the other.

How can we explain such a discrepancy? Sure, the muons are moving close to the speed of light, but we're observing them from a reference frame where we're stationary. We can measure the distance the muons travel, we can measure the time they live for, and even if we give them the benefit of the doubt and say that they're moving at (rather than near) the speed of light, they shouldn't even make it for 1 kilometer before decaying.

But this misses one of the key points of relativity! Unstable particles don't experience time as you, an external observer, measures it. They experience time according to their own onboard clocks, which will run slower the closer they move to the speed of light. Time dilates for them, which means that we will observe them living longer than 2.2 microseconds from our reference frame. The faster they move, the farther we'll see them travel.

One revolutionary aspect of relativistic motion, put forth by Einstein but previously built up by . [+] Lorentz, Fitzgerald, and others, that rapidly moving objects appeared to contract in space and dilate in time. The faster you move relative to someone at rest, the greater your lengths appear to be contracted, while the more time appears to dilate for the outside world. This picture, of relativistic mechanics, replaced the old Newtonian view of classical mechanics, and can explain the lifetime of a cosmic ray muon.

How does this work out for the muon? From its reference frame, time passes normally, so it will only live for 2.2 microseconds according to its own clocks. But it will experience reality as though it hurtles towards Earth's surface extremely close to the speed of light, causing lengths to contract in its direction of motion.

If a muon moves at 99.999% the speed of light, every 660 meters outside of its reference frame will appear as though it's just 3 meters in length. A journey of 100 km down to the surface would appear to be a journey of 450 meters in the muon's reference frame, taking up just 1.5 microseconds of time according to the muon's clock.

At high enough energies and velocities, relativity becomes important, allowing many more muons to . [+] survive than would without the effects of time dilation.

Frisch/Smith, Am. J. of Phys. 31 (5): 342–355 (1963) / Wikimedia Commons user D.H

This teaches us how to reconcile things for the muon: from our reference frame here on Earth, we see the muon travel 100 km in a timespan of about 4.5 milliseconds. This is just fine, because time is dilated for the muon and lengths are contracted for it: it sees itself as traveling 450 meters in 1.5 microseconds, and hence it can remain alive all the way down to its destination of Earth's surface.

Without the laws of relativity, this cannot be explained! But at high velocities, which correspond to high particle energies, the effects of time dilation and length contraction enable not just a few but most of the created muons to survive. This is why, even all the way down here at the surface of the Earth, one muon per second still appears to pass through your upturned, outstretched hand.

The V-shaped track in the center of the image arises from a muon decaying to an electron and two . [+] neutrinos. The high-energy track with a kink in it is evidence of a mid-air particle decay. By colliding positrons and electrons at a specific, tunable energy, muon-antimuon pairs could be produced at will. The necessary energy for making a muon/antimuon pair from high-energy positrons colliding with electrons at rest is almost identical to the energy from electron/positron collisions necessary to create a Z-boson.

The Scottish Science & Technology Roadshow

If you ever doubted relativity, it's hard to fault you: the theory itself seems so counterintuitive, and its effects are thoroughly outside the realm of our everyday experience. But there is an experimental test you can perform right at home, cheaply and with just a single day's efforts, that allow you see the effects for yourself.

You can build a cloud chamber, and if you do, you will see those muons. If you installed a magnetic field, you'd see those muon tracks curve according to their charge-to-mass ratio: you'd immediately know they weren't electrons. On rare occasion, you'd even see a muon decaying in mid-air. And, finally, if you measured their energies, you'd find that they were moving ultra-relativistically, at 99.999%+ the speed of light. If not for relativity, you wouldn't see a single muon at all.

Time dilation and length contraction are real, and the fact that muons survive, from cosmic ray showers all the way down to Earth, prove it beyond a shadow of a doubt.

## Time like spacetime interval, proper time, and time dilation

But what does “moving” mean? Surely if you’re in free fall you are justified in saying you aren’t moving, and in that special case I imagine Newton’s laws hold over small distances.

But then again you’re always at rest in your own reference frame.

I think that describes a local inertial frame, rather than “at rest.” Seem to have mixed the terms.

Being "at rest" usually is an informal way of saying that one's spatial coordinates are constant. It's a coordinate dependent statement which has no meaning until one defines the specific coordinates being used. Sometimes the coordinates are spelled out, other times they are only implied. This is somewhat sloppy, but no harm is done if both parties know and agree on which coordinates are being used. Confusion arises if both parties use different coordinates. An object "at rest" on the Earth's surface in earth-fixed coordinates is moving in barycentric coordinates, due to the rotation of the Earth, for instance.

Compare and contrast the notion of "at rest" to the different notion of "zero proper acceleration", which can be defined in an observer independent manner. An object "at rest" on the Earth's surface has a non-zero proper acceleration. An example of an object on the Earth that has zero proper acceleration would be a thrown baseball (ignoring air resistance).

Possibly you need to take a step back. Maybe the notion of being "at rest" is not what you really want to talk about.

You had been suggesting that a remote object is "at rest" if it remains at a constant distance (constant round trip light time) from a selected local object. There is a term used to characterize a set of objects that maintain a fixed distance from one another: Born Rigidity

In flat space-time, there is relative velocity -- the velocity of the one body in the [momentary, inertial] rest frame of the other. If this is zero then the objects are at rest relative to each other.

In curved space-time, the notion of relative velocity gets slippery. Non-local comparison of velocities becomes ambiguous and one needs to "parallel transport" the velocity of the one into the local frame of the other. The result can depend on the path over which this "parallel transport" is performed.

There is a different notion which might be used. Have you Googled "born rigidity"?

Hello to every body. I’ve found the attached image among my old draft papers, which I used in a conference to illustrate the so-called twin paradox:

Do you have difficulties with Italian ? In short , the test says that , in the flat spacetime of SR (no mass- energy that causes curvature) , one can go from event O to event Q in two (or more. ) ways :

1) staying at home, sitting in a very comfortable arm chair, (but let’s ignore for the moment that the Earth is not an inertial RF , from the point of view of relativity) , and just letting proper time (wristwatch time) flow. So, OQ is a piece of geodesics. Spatial coordinate x doesn’t change.
2) moving from O to Q along a curved universe line OSQ , that can be approximated by a succession of short segments , during which the speed maintains constant , and varies from piece to piece. Axes t’ , t” . are the instantaneous time axes of the MCRFs which are tangent to the curved universe line.

Then, mathematics and physics formulae don’t need to be translated. Fortunately , their language is (or should be) universal and well understood by everybody.
In the end, the result is that the integral of proper time along the curved universe line is shorter than the coordinate time from O to Q : the geodesics OQ , on the time axis of the stationary twin, is the "longest time line” . No paradox, then .

## Is there any practical proof for time dilation?

There are several direct proofs of time dilation. Extremely accurate clocks have been flown on jet aircraft. When compared to identical clocks at rest, the difference found in their respective readings has confirmed Einstein's prediction. (The clock in motion shows a slightly slower passage of time than the one at rest.)

In nature, subatomic particles called muons are created by cosmic ray interaction with the upper atmosphere. At rest, they disintegrate in about 2 x 10E-6 seconds and should not have time to reach the Earth's surface. Because they travel close to the speed of light, however, time dilation extends their life span as seen from Earth so they can be observed reaching the surface before they disintegrate.
Answered by: Paul Walorski, B.A., Part-time Physics Instructor

In October 1971, Hafele and Keating flew cesium-beam atomic clocks, initially synchronized with the atomic clock at the US Naval Observatory in Washington, D.C., around the world both eastward and westward. After each flight, they compared the time on the clocks in the aircraft to the time on the clock at the Observatory. Their experimental data agreed within error to the predicted effects of time dilation. Of course, the effects were quite small since the planes were flying nowhere near the speed of light.

A simple (well, sort of) example can be seen in experiments regarding the decay of certain particles.

For example, assume that a particle has a certain lifetime before it decays when at rest, this lifetime is known, and we shall refer to it as t. Now to test the effects of time dilation the particle is accelerated to extremely high speeds, and the resulting lifetime t' is measured by a stationary observer. To the stationary observer the particle should appear to exist for a period of time longer then it's lifetime when measured at rest (t'>t), thus confirming that time dilation does indeed occur.

'If one wishes to obtain a definite answer from Nature one must attack the question from a more general and less selfish point of view.'

## If Time does not exist what is an alternative interpretation of Time Dilation?

Einstein's theory of Special Relativity predicts that for objects travelling at a significant fraction of the speed of light time dilates. Experimental observations are in agreement with the predictions. For example ordinarily short lived particles such as Muons when at rest are observed by a stationary observer to exist for significantly longer periods when travelling at speeds approaching the speed of light.

Mathematically Speed = Distance/Time

As the speed of light is expected to be constant in any frame of reference (consequent to Maxwell’s equations) the mathematical conclusion for the increased lifespan of Muons traveling close to the speed of light would be that the values for distance and/or time have changed.

• In the observer’s inertial frame of reference time has slowed down for the Muons allowing them to live longer.

The difficulty with understanding such mathematically derived conclusions is that they are counterintuitive (which is not to say that they are wrong). Copious experimental observations illustrate clearly and consistently that clocks slow down when in motion precisely as predicted by Special relativity. Thus for example we can confidently predict that an astronaut travelling at near light speed for a year will return to Earth biologically younger than his twin brother by around thirty years.

A typical explanation of Time Dilation is that time flows at a slower rate for the astronaut than for his twin brother on Earth. The analogy of time flowing conjures up images of water moving along in a river. But as time does not appear in any real sense to be a tangible identifiable substance like water can it truly be said to be flowing at different rates? The passage of time can only be measured indirectly in terms of a perceived interval between events. The most accurate measurement of time is currently in terms of the interval between 2 quantum mechanical conditions of a Cesium 133 atom. But what really is it that we are measuring when we state that we are measuring time?

Physics defines Time as “that which is measured by clocks” that is all. There is no evidence to substantiate that time exists as part of the fabric of the universe. It is probable that human beings dreamt up the notion of time as a convenient way of 2 or more people being in the same location to share a task. For example an agreement for 2 people to meet for a hunt at sunrise on the bank of a river next to a large rock is in effect a synchronisation of the event of sunrise with 2 people and a unique geographical point on the planet. The human notion of time serves the purpose of accurately synchronising events for a species that owes much of its success to organised cooperative behaviour.

Although today we would associate sunrise with a specific time indicated on a wristwatch (or more accurately an atomic clock) there is no "known" absolute benchmark of time in any inertial frame of reference. i.e. there is no "known" universal standard time anywhere in the universe with or without the relativistic effects of speed and gravity. Significantly the sunrise over our spot on the river will never be precisely at the same local time from any one sunrise to any other sunrise as measured by an atomic clock situated by the rock. This is due in part to perpetual changes in the orbit of the Earth and in part to the uncertainty of the location and velocity of quantum particles. Quantum observations suggest that it may be impossible to predict or measure the precise local time of any event in the universe. Without any direct evidence of its existence as part of the fabric of the universe it is perhaps more useful to think of time as being an imaginary interval between 2 events.

Can there be a more intuitive way of explaining the observations predicted by special relativity?

The observation that high speed Muons last longer than Muons at rest could be interpreted in one of the following two ways:

1. Muons decay at the same rate regardless of their speed. The speed of a Muon causes time to slow down in its inertial frame of reference so that for a stationary observer for whom time is running faster a high speed Muon appears to decay more slowly than a stationary Muon. “Proper time” is the time experienced by the Muon in its inertial frame of reference being less than the time measured by the stationary observer calculated as per the following expression.
1. Muons decay at a rate that reduces according to their speed relative to a stationary observer. “Proper events” is the reduced number of decay events experienced by the Muons in their inertial frame of reference as compared with the higher number of decay events observed by the observer calculated as per the following expression.

The first interpretation founded on Special Relativity is based on the assumption that time is part of the fabric of the universe and that time literally flows at one rate for a stationary observer and at a reduced rate for the particles in motion relative to the stationary observer.

The second (alternative) interpretation assumes that time is merely a human notion and is not part of the fabric of the universe in any real sense. In this case time dilation is no longer a plausible explanation for the increased life span of high speed Muons. Since time dilation can no longer be an explanation the inference is that the high speed Muons last longer than relatively stationary Muons as a direct consequence of their relative speed.

Whilst Particles such as Muons are observed to decay into different particles it is not understood what exactly triggers the change but it is typically characterised as the spontaneous process of one elementary particle transforming into other elementary particles without any apparent external cause. There would seem to be 2 plausible interpretations:

In the first interpretation the notion that a fundamental indivisible particle may transform itself with no external influence is both counter-intuitive and inconceivably difficult to conclude from experimental observation, which is not to say that it is necessarily incorrect.

In the second interpretation, from the assumption that particle decay is influenced by other quantum events in the vicinity it follows that the rate of decay would be governed by the frequency of such quantum events.

From the same assumption that particle decay is influenced by other quantum events in the vicinity it follows that the frequency of quantum events would be governed by the values of influential properties of the quantum particles such as angular momentum.

Based on observations of particle decay being ******** in a highly predictable way according to the speed of the particles relative to a stationary observer we can further infer that the values of influential properties of quantum particles in a given inertial frame reduce with respect to the speed of the quantum particles. By considering the wave properties of a quantum particle the inference would be that the energy of the wave is reduced through dissipation over a longer distance.

An atomic clock detects an arbitrarily prescribed number of changes between 2 quantum mechanical states of Cesium 133 atoms and registers this as one second of time. A moving atomic clock detects fewer changes than a relatively stationary clock. According to Special Relativity this is due to time slowing down in the inertial frame of reference of the moving clock. However in this alternative interpretation where time is no longer considered to be a real variable the conclusion is that there are fewer quantum events occurring in the inertial frame of reference of the moving clock as a consequence of its relative inertia.

In any given inertial frame of reference the relative frequency of different types of quantum events would be expected to remain constant such that any specific measurement carried out within an inertial frame of reference would be identical to the same measurement carried out within any other inertial frame of reference. Thus for example the same values would be recorded for the average half life of a Muon at rest measured within any inertial frame of reference.

Special Relativity states that relative motion causes time to dilate. The observational evidence is that relative motion causes clocks to slow down and also causes a reduction in the frequency of all events within a moving inertial frame of reference. Thus whilst time is defined as “that which is measured by clocks” the consequences of Special Relativity do not hold clocks to be special. Although these observations can be characterised as Time dilation there is no evidence to substantiate the material existence of time and that which does not exist cannot dilate.

This alternative interpretation is founded on the same set of observations that substantiate Special Relativity but without invoking the assumed variable of time and instead substituting a relative frequency of quantum events.

#### Hartmann352

“People like us, who believe in physics, know that the distinction between past, present, and future is only a stubbornly persistent illusion”. -- Albert Einstein

Hermann Minkowski, Einstein’s one-time teacher and colleague, who gave us the classic interpretation of Einstein’s Special Theory of Relativity. Minkowski introduced the relativity concept of "proper time", the actual elapsed time between two events as measured by a clock that passes through both events. Proper time therefore depends not only on the events themselves but also on the motion of the clock between the events. By contrast, what Minkowski called "coordinate time" is the apparent time between two events as measured by a distant observer using that observer’s own method of assigning a time to an event.

An event is both a place and a time, and can be represented by a particular point in space-time, i.e. a point in space at a particular moment in time. Space-time as a whole can therefore be thought of as a collection of an infinite number of events. The complete history of a particular point in space is represented by a line in space-time (known as a world line), and the past, present and future accessible to a particular object at a particular time can be represented by a three dimensional light cone (or Minkowski space time diagram), which is defined by the limiting value of the speed of light, which intersects at the here-and-now, and through which the object’s world line runs its course.

Physicists do not regard time as “passing” or “flowing” in the old-fashioned sense, nor is time just a sequence of events which happen: both the past and the future are simply “there”, laid out as part of four-dimensional space-time, some of which we have already visited and some not yet. So, just as we are accustomed to thinking of all parts of space as existing even if we are not there to experience them, all of time (past, present and future) are also constantly in existence even if we are not able to witness them. Time does not “flow”, then, it just “is”. This view of time is consistent with the philosophical view of the eternal or the block universe theory of time*.

According to relativity, the perception of a “now”, and particularly of a “now” that moves along in time so that time appears to “flow”, therefore arises purely as a result of human consciousness and the way our brains are wired, perhaps as an evolutionary tool to help us deal with the world around us, even if it does not actually reflect the reality.

However, if time is a dimension as Rod Serling of The Twilight Zone has explained, it does not appear to be the same kind of dimension as the three dimensions of space. For example, we can choose to move through space or not, but our movement through time is inevitable, whether we like it or not, and it is often called the "arrow of time"**. In fact, we do not really move though time at all, at least not in the same way as we move through space. Also, space does not have any fundamental directionality (i.e. there is no “arrow of space”, other than the downward pull of gravity, which is actually variable in absolute terms, depending on where on Earth we are located, or whether we are out in space with no gravitational effects at all), whereas time clearly does.

With the General Theory of Relativity, the concept of space-time was further refined, when Einstein realized that perhaps gravity is not a field or force on top of space-time, but a feature of space-time itself. Thus, the space-time continuum is actually warped and curved by mass and energy, a warping that we think of as gravity, resulting in a dynamically shaped space time. In regions of very large masses, such as stars and black holes, space-time is bent or warped into a deep well by the extreme gravity of the masses, an idea often illustrated by the image of a rubber sheet distorted by the weight of a bowling ball.

Time dilation is just one consequence of the Theory of Relativity and curved space-time

Also as a result of Einstein’s work and his Special Theory of Relativity, we now know that rates of time actually run differently depending on relative motion, so that time effectively passes at different rates for different observers traveling at different speeds, an effect known as time dilation. Thus, two synchronized clocks will not necessarily stay synchronized if they move relative to each other. There is also a related effect in the spatial dimensions, known as length contraction, whereby moving bodies are actually foreshortened in the direction of their travel.

Time dilation (as well as the associated length contraction) is negligible and all but imperceptible at everyday speeds in the world around us, although it can be, and has been, measured with very sensitive instruments. However, it becomes much more pronounced as an object’s speed approaches the speed of light ( relativistic speeds). If a spaceship could travel at, say, 99% of the speed of light, a hypothetical observer looking in would see the ship’s clock moving about twice as slow as normal (i.e. coordinate time is moving twice as slow as proper time), and the astronauts inside moving around apparently in slow-motion. At 99.5% of the speed of light, the observer would see the clock moving about 10 times slower than normal. At 99.9% of the speed of light, the factor becomes about 22 times, at 99.99% 224 times, and at 99.9999% 707 times, increasing exponentially. In the largest particle accelerators currently in use we can make time slow down by 100,000 times. At the speed of light itself, were it actually possible to achieve that, time would stop completely.***

Perhaps the easiest way to think of this difficult concept is that, when an object or person moves in space-time, its movement “shares” some of its spatial movement with movement in time, in the same way as some northward movement is shared with westward movement when we travel northwest. What forces this sharing of dimensions is the invariant nature of the speed of light (slightly less than 300,000km/s), which is a fundamental constant of the universe that can never be exceeded. Thus, the slowing of time at relativistic speeds occurs, in a sense, to “protect” the inviolable cosmic speed limit (the speed of light).

It should be noted that, although a spaceship travelling at close to the speed of light would take 100,000 years to reach a distant star 100,000 light years away as judged by clocks on Earth, the astronaut in the spaceship might hardly age at all as he travels across the galaxy. This characteristic of relativistic time has therefore spawned much discussion of the possibility of time travel.

According to Einstein, then, time is relative to the observer, and more specifically to the motion of that observer. This is not to say that time is in some way capricious or random in nature – it is still governed by the laws of physics and entirely predictable in its manifestations, it is just not absolute and universal as Newton thought (see the section on Absolute Time), and things are not quite as simple and straightforward as he had believed.

One casualty of the Theory of Relativity is the notion of simultaneity, the property of two events happening at the same time in a particular frame of reference. According to relativistic physics, simultaneity is NOT an absolute property between events, as had always been taken for granted by Newtonian physics. Thus, what is simultaneous in one frame of reference will not necessarily be simultaneous in another. For objects moving at normal everyday speeds, the effect is small and can generally be ignored (so that simultaneity CAN normally be treated as an absolute property) but when objects approach relativistic speeds (close to the speed of light) with respect to one another, such intuitive relationships can no longer be assumed.

## Time Dilation and Particle Decay - Astronomy

Galaxy’s tug changes particles’ ways
GEMMA LAVENDER
ASTRONOMY NOW
Posted: 15 July 2011

The reasoning behind why different amounts of matter, and associate antimatter, has survived the birth of our Universe may just have been solved by a physicist of the University of Warwick.

The two forms differ in the sense that matter is composed of regular particles, whereas antimatter is made up of antiparticles.

Mark Hadley of the university’s Department of Physics, believes that he has found a testable explanation for what is known as apparent Charge Parity (CP) violation where parity is preserved, but the violation is made all the more of a reasonable explanation for the divide between matter and antimatter. CP violation is  a particle physics theory where the laws of physics are violated if a particle were interchanged with its antiparticle (a particle which has the same mass but opposite charge) in charge conjugation symmetry, and left and right are swapped in what is referred to as (parity symmetry). The violation was uncovered in 1964 via the decays of neutral kaons, a particle understood to contain a strange quark paired with an up or down antiquark, resulting in its discoverers, James Cronin and Val Fitch, obtaining a Nobel Prize in Physics in 1980.

Artist’s impression of the frame dragging effect of a galaxy on a grid with particle decay trails superimposed on top. Image: University of Warwick/Mark A Garlick.

In recent years, experimental observations of kaons and B-mesons (a subatomic particle composed of a bottom quark or bottom antiquark) have illustrated notable differences in how their matter and antimatter versions decay. With this, CP is therefore violated, presenting itself as an awkward anomaly for some scientists but simultaneously revealing itself as useful in attempting to explain why more matter than antimatter appears to have survived the turbulence that was the birth of our Universe.

However, Hadley, whose research paper has been published in Europhysics Letters, believes that while trying to solve this problem, scientists have neglected an important factor the impact of the rotation of our Galaxy and its influence on how subatomic particles break down. “Nature is fundamentally asymmetric according to the accepted views of particle physics. We watch what elementary particles do and we find that some patterns of interactions and decays exist, but the mirror image pattern is absent and this means that there is a clear left-right asymmetry in weak interactions and a much smaller CP violation in kaon systems,” says Hadley. “These have been measured but never explained.”

“This research suggests that the experimental results in our laboratories are a consequence of galactic rotation twisting our local space-time,” he says. “If that is shown to be correct then nature would be fundamentally symmetric after all. This radical prediction is testable with the data that has already been collected at CERN and BaBar by looking for results that are skewed in the direction that the galaxy rotates.” Although it is considered easy to overlook the bigger picture and neglect the effects of something as large as our Galaxy’s effects on us when we have the Sun close by, Hadley believes that the effect generated by a massive spinning body is highly important.

An artist’s conception of the Milky Way galaxy, whose twisting effect is said to be responsible for the differing ways in which the particles decay. Image: Nick Risinger.

It is this effect that generates a speed and angular momentum that drags the reference frame of local space and time, twisting its shape and generating time dilation effects. Due to its size in comparison to the Earth, the Milky Way galaxy has a tugging effect so strong that it is a million times greater than that caused by the spin of our home planet. The frame dragging of the whole Galaxy, according to Hadley, explains the observations of CP violation in the decay of B mesons where the key difference between the break-up of matter and antimatter versions of the same particle lies in the variation of the decay rates. The combination of the decay rates combined, however, equal the same total for both matter and antimatter types of the same particle which also retain the same structure, becoming mirror images of each other.

Expecting these particles to begin in this state is one expectation, but how they end up after their decay is quite another and it is the tug of the galactic frame which is held responsible. The twisting causes the different structures in each particle to experience different levels of time dilation and therefore decay in different ways, although overall variation of the varying levels of time dilation evens out when considering particles individually and CP violation disappears conserving parity. A massive advantage to the theory is that it can undergo tests and this up-side has not gone unnoticed – large arrays of data are very real, illustrating apparent CP violation in some decays, which can be re-examined to find a pattern that is aligned with the rotation of the galaxy.

While Hadley’s paper, entitled The asymmetric Kerr metric as a source of CP violation , only explains how galactic frame dragging could explain the experimental observations of CP violation, it provides a lead to scientists who regard the violation theory as a useful tool to explain the separation of matter and antimatter at the birth of our Universe.

## Elements of Physics (111)

The cloud chamber was the first tool physicists had that enabled them to see the elementary particles that resulted from radioactive decay. It is a fundamentally simple device containing vapor that is just ready to make a cloud. All it needs is something to condense on, and that may be the ions produced when an energetic charged particle zips through the gas. The resulting cloud, much like the contrail of a jet airplane at high altitude, is the track of the particle through the chamber.

The first one was made by Charles Wilson in 1911. You can see it at the science museum at the Cavendish Laboratory.

It makes a saturated vapor in water suddenly removing air from the chamber, so it cannot show tracks for a long period but it works very well for snapshots of the cosmic rays. It is the predecessor of the liquid bubble chamber that was the workhorse tool of high energy physics in the middle of the last century.

At about the same time, Victor Hess carried electrometers, devices that measure the rate of ion production in a gas, on a ballon flight up to 5300 meters, and he found that there were more ions being made at higher altitude than down on Earth's surface. The source of the ionization must therefore be above Earth's atmosphere, and the term "cosmic ray" was adopted some years later. Hess received the Nobel Prize for his discovery.

The primary cosmic ray is typically an energetic proton, or occasionally a heavy ion, from the Sun, a supernova, or some other distant source in the galaxy. When it strikes an atom in the atmosphere, among other products it produces μ-mesons, and it is these muons that are responsible for the ionization and tracks we see in cloud chambers at lower altitude. In our study next week we will see that these particles are very similar to electrons, except more they are more massive and they are unstable. A muon will decay if left on its own primarily by this process

which is to say it turns into an electron and a pair of neutrinos. The lifetime of the muon before it does this is 2.2 μs (a microsecond is 10 -6 second, a millionth of a second). If a muon goes through the chamber it ionizes the gas, but if it stops, about 2 microseconds later it produces an electron that also ionizes the gas. (The neutrinos hardly interact, so we do not see them at all.)

#### The problem

If muons only live for 2.2 microseconds, how is it they get down to Earth's surface? In that short a time, at the speed of light, the fastest speed they can go, they will travel

where (Delta t) is the elapsed time and (c) is the speed of light. That works out to

meters or 0.66 kilometers. The upper part of Earth's atmosphere is of the order of 100 km away, and even at 10 km it is too far. How do cosmic rays make it down to the surface?

#### Observations

recorded at the astronomical observatory at Pic du Midi in France, 2.9 km above sea level. The chamber is 45×45 cm and about 5 cm high. Most of the tracks it detects are from horizontally traveling particles, other experiments have shown are only about 5% of the actual flux that is mostly vertical. This video is recorded in real time, so watch some of it and see if you can estimate how many new tracks there are per second. You may have to limit your attention to short interval and play it frame by frame to count the events. It is not an easy task. It won't matter if you miss a few. The best way is to examine the video for several seconds, then divide your count by the time.

1. Are all the tracks the same, or are some different?

2. How many did you count per second?

3. What is the volume of the chamber, in cm 3 ? (Hint: simply multiply length by width by height in cm.)

4. How many were there per cm 3 per s? Estimate the error in your measurement by trying this in a few different time intervals and comparing your results. If you multiply by 20, that's the number of vertically traveling ones, so the total will be approximately 20× the number you actually count. Most (but not all) are muons. You can look at other videos too. The results depend on the altitude, because the particles are absorbed by air, and on the efficiency of the chambers that depends on their design and how saturated the vapor is.

5. In the same units of cm 3 estimate your own volume. Yes, that's hard to do accurately. One way is by displacement. If you put something into a liquid, the volume of the liquid displaced is the volume of the object submersed. We're talking about bathtubs here! Or, you can take your weight and divide by the average human density which is about 985 kg/m 3 , If you only know your weight in pounds, then use 1 kg = 2.2 lb. A cubic meter is 10 6 cubic centimeters.

6. If you were at Pic du Midi, how many cosmic ray muons would go through you, per second?

Your exposure to cosmic rays, which cause mutations and possibly cancer, is higher the higher the altitude. Pilots and flight attendants may be classified as radiation workers for health purposes because of this risk, and they receive an average annual dose of about 3 milli-sieverts (mSv). The sievert (Sv) is unit of ionizing radiation energy that is equivalent to 1 joule of energy absorbed per kilogram per meter 2 per second. That is thought to convey a 5.5% risk of eventually developing cancer. Flying at cruise altitude delivers a dose of about 2.7 μSv/h, but this exposure is cumulative which is why there is a risk to high flyers. (A CT scan delivers a dose of around 30 mSv, and a banana which has naturally radioactive K (potassium) provides a does of around 0.1 μSv.)

#### Time dilation

So how is it that muons survive to reach us at sea level? Several of the video links in this activity will help to understand that the reason, as we see it, is time dilation. The muons are traveling at nearly the speed of light. This causes their "clock", i.e. their lifetime or decay rate, to appear to us to run slowly. The detection of muons at sea level is a dramatic demonstration of special relativity. Let's do the math.

The formula for time dilation is that an interval (Delta t_0 ) seen by the moving clock (the muon, in this case), is measured by a stationary observer (us, in this case) to be

(Delta t = Delta t_0 / sqrt <1 - v^2/c^2>)

7. Assuming that the muons are traveling at 0.98 times the speed of light (something we can measure, but don't in this experiment), what would be the apparent lifetime of the muon to us?

Now for the real challenge. To the muon, its lifetime is not altered by its speed. It sees the atmosphere of Earth going by, and still gets to the ground. Why is that?