## 2015-12-06

### Relativity

I am going to stick my neck out here and try and discuss something that is on the edge of my knowledge. There are a couple of areas I know are just beyond my grasp...

One is the maths of public key encryption and sadly I had this explained and I remember getting it properly for a couple of days and losing it - that must be what Alzheimer's feels like. To be clear I do understand all of the principles and implications, it is just the maths that is that far beyond my everyday usage that I cannot retain it.

The other is relativity - mostly I get this but it is the time dilation that has me stuck, so I'll explain the bits I do grasp and where I fail. I am not really expecting a lot of answers - I will get some, and they will either be maths over my head or examples I don't find convincing. There was a recent case of some kid getting a prize for explaining relativity but the explanation had holes in it as I saw it, big holes.

So far I have grasped that gravity and acceleration are the same. This seems actually quite intuitive to me - in that they feel the same. Free fall is the same as simply travelling through space and zero gravity. Gravity at 1G on the surface of the Earth feels the same as 1G of acceleration in a rocket in space. It all fits and make perfect sense to me that they are the same.

Also, the fact that in gravity you experience time dilation - time goes more slowly compared to this not in gravity. The effect of a gravitational force is seen the same for all observers. If I am accelerating at 1G, people observing can see that, and so can I, and we all agree. The same as if I am sat on a planet with 1G force, every observer agrees. The idea that there is a time dilation due to gravity (or acceleration) is fine by me. At 1G time is slower than at 0G, and faster than at 2G. All observers see the same. Everything is fine. Let's accept that.

The other effect is speed - which is always relative. If two people are travelling at a speed away from each other, each see the other as going more slowly. This is an effect that I would see as logically an "illusion", in the same way the seeing someone at a distance they look smaller, but you look smaller to them - a paradox. If ever they change to coming towards you, you see them going faster than you, and they see you going faster to them. Ultimately this effect cancels out when you meet and have no time dilation that has accumulatively between you as a result. Whether the effect is "real" or an "illusion" is academic.

So far so good.

The problem I have is that there seems to be some effects that are speed related and do not fall in to either of the above.

The idea that someone whizzing off in a space ship and coming back later will appear to have some measurable cumulative time dilation effect. Now, this makes sense if they have some acceleration involved either way as that would slow their time compared to the person that had no acceleration and was left behind. But is that the only effect?

Apparently the ISS in orbit had two effects, one is due to lower gravity so faster time, and one is a speed effect and slower time, and they do not cancel out quite. It is the speed effect I cannot get my head around as surely to an observer on Earth they spend half their time travelling away (and hence seem slower) and half their time travelling towards (and hence seem faster), but the net effect of speed is always zero, only gravity ultimately has a cumulative effect.

So that is where I am lost - I hope one day to grasp it, and blog it.

1. Ah, no - imagine observing the ISS from the centre of the Earth, it is always the same distance away BUT it has speed. It's speed is always the same. It's velocity (vector) is ever-changing as Earth's gravity makes it orbit. A change in velocity is aceleration (even though speed, the scalar, stays the same)

1. As I say, dilation due to acceleration is fine by me

2. Also a body in orbit is the same as a body in free-fall (just taking the long route) as it experiences zero net acceleration.

3. The ISS is under constant acceleration and continuously changing velocity (at any instant its' velocity is entirely tangential to the Earth). The velocity of the ISS is such that it would continue heading straight in its' current direction of travel should Earth spontaneously cease to exist.

It undergoes gravitational pull at 90 degrees to that direction of travel - that is, a gravitational pull towards the Earth - which causes the orbit to curve and hence form a circle (or more generally an ellipse)

Therefore, for an observer at the center of the Earth the ISS is continually accelerating towards them! At no point does the ISS accelerate away!

This is one area where calculating things in Euclidean coordinates is misleading - taking a Eucliean average of the acceleration vectors, yes, they null - but in Polar coordinates they do not. Polar coordinates tell the correct story here.

From the perspective of somebody on the surface of the Earth this no longer applies - the ISS does sometimes decelerate away, because the radius of the Earth is close to the radius of the orbit of the ISS - but the ISS spends more time accelerating towards them than away, so time dilation effects still apply

2. A body in orbit and a body in free fall are both at 0G but they are both accelerating
They both have a changing vector
Only one has a changing scalar
Energy remains the same in both

3. Does this help?
https://www.quora.com/Why-does-time-slow-down-as-you-approach-the-speed-of-light

4. I'm a bit tired, but here goes (and it's been a long time) - "some acceleration involved either way as that would slow their time compared to the person that had no acceleration".

Differences in Gravity affects time dilation. Acceleration from an "inertial frame of reference" (man in a lift) same as Gravity. Time dilation also depends on 'relative' **velocity** (relative to what?). Don't confuse Acceleration/Gravity with relative velocity...

Your relative velocity is just the speed difference between Airplane in direction A & Airplane in Direction B, with C being a stationary rock between the two. A/C and B/C see half the time dilation as A/B.

I think it would help to separate general vs. special relativity -https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment

I'm off to bed.

5. I find these guys pretty good: https://www.youtube.com/channel/UC7_gcs09iThXybpVgjHZ_7g

What you're probably looking at is the contraction of spacetime along the path of travel. That is, as you travel faster, you travel less distance as well - by going fast enough the distance between any two points in space contracts towards zero. By my equally weak understanding anyway.

6. edit - Plane A flying west, Plane B flying east....

7. The slowdown of the external universe as you approach the speed of light with reference to it is not an illusion, if by "illusion" you mean anything that can be shown to have not happened. You say: "Ultimately this effect cancels out when you meet and have no time dilation that has accumulatively between you as a result.", but this is not true!

The slowdown of the external universe applies *just as strongly* and is just as real a physical effect to the person on the spacecraft looking back at you, and to you looking at events on the spacecraft: each will see the other's clocks running slow, and *each will be right*. There is no experiment it is possible to perform that will show anything else. However -- this is a special-relativistic result, and special relativity only applies to non-accelerated frames. So the effect will not cancel out when you meet -- you'll flash past each other at near lightspeed, and each will *still* see the other's clocks running slow, and as you move away from each other the slow clocks will continue. You can imagine what's happening here is that you're looking at the other guy's clock at two consecutive intervals, but because the clock is moving so fast with reference to you, light has to travel further to get to you at one instant than it does at the other, throwing off your image of the clock such that it seems as if it has advanced less than it should, because the light had to travel a different distance and thus took a different amount of time to reach you: but that image is the best you can get. It is, as far as you can describe these things, your local truth. The other party's local truth will be different.

If one party accelerates or decelerates, you're suddenly in the realm of general relativity. You can imagine what's going on here by thinking of a light beam passing from a stationary source (you on earth, say) to a spacecraft with me on it moving away at near lightspeed, making my Internet latency simply terrible. Every second you send a pulse, but because the spacecraft is moving away rapidly it sees these pulses arriving much more slowly: that's time dilation. But now, imagine the spacecraft starts to slow down until it is at rest with reference to Earth. All those light pulses that have been chasing it will start to catch up faster, until (when it is stationary) they are arriving as fast as they are leaving on Earth. If I then accelerate towards Earth again to try to get my ping times down, they will start to catch up *faster*, arriving with less than a second between them.

But those pulses are my idea of your local clock -- my idea of the passage of time on Earth. In a very real sense they *are* the passage of time on Earth, in my reference frame. As I start to accelerate towards you -- having gone away from you before now -- time on Earth *speeds up* with reference to me, rather than dilating -- and when I decelerate at Earth (not wanting to crash into it in the pursuit of better WoW latencies), it slows down again. If you do the maths the effect of this is that we do see a difference in durations between Earth and the spaceship, and it's always positive (more time has always passed for the non-accelerated body), but it's not as big a difference as you'd expect if the spaceship had looked back at Earth at its maximum distance and looked at the clock difference then.

(more in the next post, this is 'too long', bah.)

8. (continued from previous post.)

So far, so obvious doppler effect. But now... gravity and acceleration are the same thing. When I was accelerating back towards you at the end of my trip, I was in a (self-created) gravitational field, where down was pointed away from the Earth: and because all motion is relative, it is equally valid to describe me as accelerating with respect to the universe, and *the entire universe* as accelerating with respect to *me*. So the entire universe was, from my perspective, in a gravitational field, where 'down' pointed towards the back of my spacecraft (away from the Earth, since I was accelerating towards it).

So, what effect did that field have? From a couple of paragraphs up, we already know what we saw: the light pulses were arriving ever more rapidly as I slowed to relative rest and started moving towards you again: i.e. it looked as if time passed ever more slowly for me with respect to the half of the external universe with Earth in it, that part that was, from my perspective, 'up', and thus in a weaker field! (Furthermore, when I was accelerating away from Earth, the opposite was true: 'down' was pointing towards the Earth, so from my perspective the Earth was in a gravitational field caused by my acceleration: so time slowed down for the Earth with respect to me, precisely replicating what you'd see from those clock pulses as I accelerated away at the start and the clock pulses grew ever further apart.)

It all really does work, and what's particularly amazing about SR and GR in particular is that the thought experiments explain so much of it without ever having to get into Minkowski spacetime and metric tensors and time being negative to space and all the formalisms. You could never get this sort of mental picture with quantum mechanics.

9. "If ever they change to coming towards you, you see them going faster than you, and they see you going faster to them."

No, your clocks will still appear to be ticking slower to them and their clocks will still appear to be ticking slower to you. You do /not/ experience "time constriction" when things are travelling towards you.

1. If that were true, which of the two clocks seems to have actually lost time when the two people eventually meet - and why?

2. Going to keep this very short.

There are only two possible universes - one where the (maximum) speed of information is infinite and one where it's finite.

If the maximum speed of information is infinite then all clocks can be synchronized everywhere - just have a "reference clock" that sends out ticks at infinite speed to everywhere in the universe - everyone agrees on the time. (our universe doesn't look like this as far as we can tell based on all experiments to date)

But if the speed of information is finite then there's two different ways to synchronize clocks:

a) bring them together. While they're together you can synchronize them
b) transmit a signal from one to another and do a calculation to (try to) set them to the same time and ticking at the same speed.

So the "twins paradox" is resolved because if you synchronize two clocks at the same point that are travelling at different speedsthen either you i) have to do a calculation including sending signals from one to the other to determine which one is running faster (and the calculation itself is affected by the "shape of the universe") or
ii) if you bring them back together again then at least one must have undergone acceleration along part of its journey and the one that has undergone acceleration runs slower.

Gravity causes an acceleration therefore clocks in a gravity well run slower. Once the curvature of space is enough (event horizon) clocks stop.

3. It depends entirely on how they meet.

If they *collide* at their original speeds, they'll probably be reduced to their component quarks before you can write down their displays. (It might be an interesting experiment for the LHC, though.)

If one of them accelerates to match velocity with the other, it will slow down to match the observed rate of the other clock.

If both of the accelerate to some intermediate velocity, then both of them will slow down in fact, but appear to speed up when observed by the other. Again, they will have matching rates when they rendezvous.

The Twin Paradox is a useful mental crutch here. In the traditional form, one twin flies off at a large fraction of the speed of light, and returns some time later, while the other twin remains more-or-less stationary at home. In such a case, the travelling twin will have aged less. But if both twins travel (in opposite directions, say), they'll both age less relative to others who stayed at home, and by the same factor (if they accelerated and travelled at the same rates for the same lengths of time).

4. "If that were true, which of the two clocks seems to have actually lost time when the two people eventually meet - and why?"

If they are travelling at constant velocity towards each other, they have never been able to synchronise their watches, as this is the first time they ever meet. Therefore, the question as to "which of the two clocks seems to have actually lost time" makes no sense.

5. Two people start together, they travel away from each other and each see the other's clock going slower. Then they travel towards each other. Now, you say they each see their clocks going slower still. When they meet they must each think the other has "lost time" - are their clocks no longer in sync?

6. In order to start together and then meet again, both must undergo acceleration (either by turning around and going the opposite way or by going in a circle).　This changes their overall observation and they will not think the other has lost time relative to them (although I admit I cannot fully explain exactly why they would think that way - but it is the fact that they know they are have been accelerating that changes their perception).

7. Special relativity - which is what we're talking about here -- depends on inertial reference frames -- there is no acceleration and therefore you can never go out and return.

Consider two twins who travel apart at x% of the speed of light (maximum speed of information).

Relavitity basically says that there is no preferred reference frame, therefore everything has to look the same whichever twin we decide is moving and whichever twin is stationary.

For each twin the other twin is moving at speed x and we will assume there's a "time dilation factor" of D for that twin relative to the other one.

We'll work in the frame of reference of twin A. i.e. do the calculations as though we stay with twin A.

The two twins agree as they pass each other that they will each send a message in 1 years time as they perceive it to the other twin. Each twin agrees when they receive that message they will then send a message back saying how old they were when they received it and so on.

Relativity says that both twins must receive the messages from the other twin when they are the same age.

From twin A PoV. After 1 year twin A sends the message to twin B saying "one year has elapsed". As far as A is concerned, this message will arrive at twin B at time 1/(1-x) (infinite sum of geometric progression). Because twin B "suffers" time dilation the age of B when they receive the message will be D/(1-x).

From twin A PoV twin B sends a message saying 1 year has elapsed after 1/D years. The separation will be x/D so twin A will receive the message at time (1+x)/D. So twin A will send the second message when he is (1+x)/D years old.

Relativity says that these two messages must be the same - so (1+x)/D = D/(1-x) => D=sqrt(1-x*x)

Now here's when it starts getting fun.

Let's assume that the twins are separating at sqrt(3/4) of the speed of light. (Time dilation factor of 0.5)

From twin A PoV when he sends the message to twin B saying one year has elapsed it will take 6.46 years to catch up with B - total elapsed time will be 7.46 years from twin A POV. But the time dilation factor for twin B will be 0.5 so twin B will send a message back saying "I'm 3.73 years old" and that message will take 6.46 years to get back to A (at which point A will be 13.93 years old)

Again, from twin A PoV twin B has a time dilation factor of 0.5 so twin B will send a message back to A saying "one year has elapsed" after 2 of A's years. And B will now be 1.73 light years away so A will receive it after 3.73 years. So A will then send the message back to B saying "I got your message when I was 3.73" - exactly the same age as B was when he got A's message.

Now B is racing away from A. From A PoV his response back to B won't arrive until A is 27.86 years old. However, B's time dilation means that B will only be 13.93 years old when he receives it - exactly the same age as A was when he received B's response.

And so the next message they each send is "I'm 13.93 years old".

There's nothing we can do and no experiment we can do to determine which twin is "older". From each twin's PoV the other ages slower (when twin A sent his first message to B he knew it would only arrive after a total of 7.46 years but twin B claimed he was only 3.73 when he received it)

If one twin turns around and heads back to the other twin then that twin will have undergone acceleration and will actually age less than the twin that doesn't accelerate. If they both turn around and come back together then they will both continue to agree that they are the same age - but will age less than someone who stayed in the CoM frame.

10. The best explanation I heard was:

Imagine a clock that works by having two mirrors that face each other. A pulse of light bounces between the mirrors, and (since the speed of light is a constant) takes a constant amount of time to bounce between the mirrors. If the mirrors are 1 meter apart, then you can count 299,792,458 bounces and that's one second.

Now put that clock on a train so the light beam is bouncing up and down. If the train is stationary, then an observer on the train and to one standing on the ground next to the tracks both see the clock ticking at the same rate.

If the train is moving at 100mph, then the observer on the train still sees the light beam travelling straight up and down at the speed of light. But an observer standing on the ground sees the light beam travelling at an angle - the mirrors are moving so the light beam has to be moving slightly in the direction the train is travelling as well as up and down. This means the light beam has to cover a slightly longer distance between the two mirrors. (The velocities are at right angles so you can use Pythagoras' Theorem to calculate how much longer). The light is still moving at the speed of light. This means that the observer on the ground sees the clock running slightly slow.

Since both readings are valid, we conclude that time slows down as you travel faster.

11. There are two concepts that I'm pretty sure you (and everyone else here) are missing. They might not close the gap entirely, but I think they're essential.

First is that there is no concept of *simultaneity* in relativity - only *concurrency*. This semantic distinction also crops up in multithreaded programming, and for the same reasons. The basic distinction is:

- When two events are simultaneous, you cannot truthfully state, from *any* point of view, that either occurs before the other.

- When it is impossible to determine, independently of viewpoint, whether one event comes before or after another, the two occur *concurrently*.

- When two events occur concurrently, either could be said to occur before the other by choosing a suitable viewpoint to observe them. (Relativistic jargon is "reference frame" for viewpoint.)

In relativity, *no* two events are simultaneous unless they occur in the same place as each other as well as at the same time. Additionally, two events are *always* concurrent if they occur outside each others' "light cones".

The "light cone" of an event is the (four-dimensional) volume of spacetime which light from that event can reach, or from which light can reach the event, due to the finite speed of light and possibly the curvature of space. The lightcone is not, however, blocked by opaque objects as real light would be.

In short, the lightcone represents the limits of causality.

Note also that two events that are outside each others' lightcones (ie. concurrent) can be observed together from a single viewpoint, which is *inside* both lightcones. However, depending on the viewpoint, they may *appear* to occur in either order, or even simultaneously.

The concept of lightcones leads to the idea of "spacelike" and "timelike" curves in spacetime. A timelike curve describes the motion of an object in space over time. A spacelike curve is what we use to describe distance. However, they exist in the same spacetime, and are differentiated only by whether they lie entirely inside or entirely outside the lightcone (respectively).

12. You are muddling the time dilation effects of velocity and gravity. These are entirely separate effects, so you need to look at them separately.

Time dilation due to velocity is best explained with the photon clock, a photon bouncing between two mirrors. If the mirrors are 1 meter apart, then an observer stationary relative to the mirrors will count 299,792,458 bounces per second. An observer not stationary to the mirrors will count fewer bounces per second according to his own reference clock which is stationary relative to him. Therefore, to the observer moving relative to the mirrors, time appears to be running more slowly at the mirrors. The reverse observation would have the same result.

Any observer moving at a constant velocity (i.e. not experiencing acceleration) will always observe time to run slower at an object moving relative to him, regardless of whether the object is moving towards him, away from him, past him or around him.

This resolves the space station issue. From an observer on earth's point of view, the space station is moving. If the earthbound observer could observe a photon clock on the space station, he would see fewer than 299,792,458 bounces per second according to his earth clock. The acceleration of the space station due to being in orbit has nothing to do with this, and that's where your first flaw is in your original post. As a result, the space station accumulates time dilation from the earth's observer's point of view (minus some time contraction due to less gravity on the space station, but this is an entirely separate effect).

As for the twin paradox, why does one twin come back older than the other, when both observe the other's time as running slowly? This is because one twin undergoes acceleration and the other does not. But it is _NOT_ the time dilation effect due to acceleration that makes the difference. That was the second flaw is in your original post. It is because the travelling, in order to return to earth, must undergo some acceleration (either by turning around and going the opposite way or by going in a circle), and therefore he has not travelled at constant speed in the reference frame of the earth, whereas the resting twin has, so therefore when then the twins meet again there is an actual difference in the time that has passed for each of them.

13. Thanks for your public contributions to the cause of common sense.

I once had trouble with the idea of a cumulative time-dilation effect because I thought that special relativity was an APPARENT effect due to the finite propagation speed of light, i.e. once you meet the traveller in the same place your clocks have to agree again because light takes no time to pass between you any more.

This is wrong: special-relativistic time dilation is what you measure AFTER you've compensated your observations for the finite speed of light.

Imagine that someone's travelling directly away from you at a speed near 'c': the images you see with your light-based telescope will be increasingly delayed as the traveller gets more distant from you and the light takes longer to travel. This will make the traveller appear to move away more slowly than they actually are.

If you do the calculations and allow for this increasing lightspeed delay, though, you can calculate where the traveller "is" from moment to moment and get an "objective" picture of their relative motion.

It's THAT picture that obeys the transformations of special relativity, not the observed image.

In some special cases (e.g. a cube-shaped object shooting past you at relativistic speed), the effect of finite lightspeed can appear to counteract special-relativistic effects. In the cube's case, it will appear not as it is (foreshortened) but tilted instead.

PS: the crux of the twin paradox is that whichever twin does more ACCELERATING during the overall journey is the one that experiences less time passing. In special relativity, acceleration is a more objective concept than velocity - the twins' clocks slip apart during the periods of acceleration and NOT during the intervening time.

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