Understanding Time Dilation: How Fast Do You Have to Go?

[DrGreg]

I do not think your tables are honest. Your first table showed the turn around took place when the moving clock has accumulated 60 minutes, 60 vs 40 (40 was the illusion in the telescope. Actual number was 60 due to 20 minutes delay). Your second table showed the turn around took place when the moving clock has accumulated 90 minutes, 90 vs 60 (60 is the illusion in the telescope. Actual number is 80 due to 20 minutes delay.). This was the point I could not understand. I did not find any solution to it in your post.

The turn around occurs when Alice reaches (m), which is at 16:00 according to her own clock. This is something everybody agrees upon. Alice knows it because she is there and looks at her own clock when it happens. Bob does not know that it has happened immediately but has to wait until light has traveled from (m) to (a). When this has happened, Bob can see that Alice has turned round and he can see that her clock was showing 16:00 when it happened.

If you accept that the Doppler factor of 2/3 applies to both Bob's view of Alice and Alice's view of Bob, then you have to accept that the red rows of both tables are correct.

My understanding of your original tables was, it was a two hour return journey, one hour each way.

...according to Alice.

I do not understand how could the turn around take place when one clock has accumulated 90 minutes.

It doesn't. Bob sees the turn around when his clock shows 13:30, but the turn around has already taken place some time earlier. It takes time for light to travel from Alice at (m) to Bob at (a).

I did not challenge the truthfulness of the equal row, nor the definition of simultaneity. What I did challenge was your consistency. You changed rules of the game by prolonging the one way journey to 90 minutes which made the equal row disappear, and as a result the turn around took place 30 minutes later.

Alice's journey appears to Bob to be 90 minutes because of the delay in light traveling from Alice to Bob. The 2/3 factor applies: a journey of 60 minutes by Alice's clock looks to Bob like a journey of 60 / (2/3) = 90 minutes. Alice's journey appears to Alice to be 60 minutes because she sees no delay on her own clock. The return trip journey is not symmetrical -- it is Alice who turns round, not Bob. The event of her turning round is experienced directly by Alice but can only be observed remotely, after a delay, by Bob.

Although you changed the time on Alice clock to be 15:00 hours at start, but the equal row was still there in the first table above, 13:40 and 16:40, accumulation of 100 minutes on each clock, or 1:40 hours, meaning light is instantaneous. How could it be so?

I answered that in my last post. Once clocks have been separated you cannot apply the time on one clock to events that occur at the position of the other clock.

Think of a clock like the trip-meter on a car. If you zero the trip-meters on two cars, and then the two cars follow each other along the same road, at the end both trip-meters show the same mileage. But if the two cars follow different routes, when they later meet, the mileage may be different.

For example suppose you and I meet in New York and zero the trip-meters on our cars. You drive directly to Los Angeles. I drive to New Orleans first, then I drive on to Los Angeles. When you and I meet again in Los Angeles, I have driven further than you. Our trip-meters show different numbers even though we are both in the same place.

Clocks behave the same way. If two clocks follow two different routes through space-time, they may show different accumulated times when they are brought back together again.

I had an inkling that people cannot make the time dilation idea stand unless they contradict themselves, such as the equal row. I believe it was all an illusion. You said so in the beginning; you knew it was.

This is what I said in post #68:

From Alice’s point of view, it looks as though Bob’s clock is ticking at 2/3 of the rate of her own clock. Of course, this is an illusion – it is nothing more or less than the Doppler effect, caused by the delay of the light signals.

The illusion would be if Alice believed that Bob's clock really was ticking at 2/3 the rate. It is still true that what she sees (after a delay) is Bob's clock's image which ticks at 2/3 the rate of the image of her own clock.

 

[JesseM]

Alice's journey appears to Bob to be 90 minutes because of the delay in light traveling from Alice to Bob.

Just so Sam doesn't get confused here and think it's all an illusion due to light-signal delays, it should be pointed out that that's not the only reason it appears to be 90 minutes--there is also genuine time dilation of Alice's clock happening in Bob's frame, the relativistic doppler shift equation takes this into account (that's why it's different from the non-relativistic doppler shift equation). DrGreg, what was the relative velocity between Alice and Bob in this problem?

 

[jtbell]

what was the relative velocity between Alice and Bob in this problem?

Inverting the relativistic Dopper effect equation gives me
$$\frac{v}{c} = \frac {1-R^2}{1+R^2}$$
where $R = 2/3$, so $v = 0.3846c$, or $\gamma = 1.0833$.

 

[Sam Woole]

You're misunderstanding the second table. The second table shows that Alice's clock read 16:00 at the time of the turnaround, but the light from that event didn't reach Bob until his clock read 13:30. The first table, on the other hand, says that Alice's clock read 16:00 at the time of the turnaround, and that at that moment she was seeing light from Bob's clock which read 12:40. So in both tables it's true that the turnaround happened at 16:00 according to Alice's clock, so both tables say the moving clock accumulated 60 minutes.

No, I was not misunderstanding the second table.
Let me quote your words: "The first table, on the other hand, says that Alice's clock read 16:00 at the time of the turnaround, and that at that moment she was seeing light from Bob's clock which read 12:40." Yes, but at this juncture what would Bob see on Alice's clock? Of course he would not see 16:00, but 15:40 on Alice's clock (16:00 minus 20 minutes delay). I think this positively means that, the turnaround took place when Bob saw 15:40 on Alice's clock. Therefore from my viewpoint, the second table should be revised accordingly.

 

[JesseM]

No, I was not misunderstanding the second table.
Let me quote your words: "The first table, on the other hand, says that Alice's clock read 16:00 at the time of the turnaround, and that at that moment she was seeing light from Bob's clock which read 12:40." Yes, but at this juncture what would Bob see on Alice's clock?

What do you mean by "at this juncture"? Do you mean, "at the same time Alice's clock read 16:00"? The problem is, again, that different frames define simultaneity differently, so they will have different answers to what Bob's clock read at the same time that Alice's read 16:00. Since $$\gamma = 1.0833$$ in this example, then in Alice's frame Bob's clock is running slower than hers by a factor of 1.0833, so after her clock has accumulated 60 minutes, Bob's clock has only accumulated 60/1.0833 = 55.386 minutes, meaning that his clock reads 12:55.386 at that moment. In Bob's frame, it is Alice's clock that is running slow by a factor of 1.0833, so when her clock has accumulated 60 minutes, his clock has accumulated 60*1.0833 = 64.998 minutes, meaning in his frame, his clock reads 13:04.998 minutes at the same moment her clock has accumulated 60 minutes.

Of course he would not see 16:00, but 15:40 on Alice's clock (16:00 minus 20 minutes delay).

Delay times don't change the readings on a clock right next to an event! If Alice pushes the button to turn around, and a clock on her dashboard right next to that button reads 16:00, then all observers, no matter how far away, will see her clock reading 16:00 when she pushes the button when they look through their telescope. Imagine that Alice wrote down the time on a piece of paper after pushing the button--do you imagine that different observers would see her write down different numbers depending on the delay time?

Maybe you're not talking about what time Bob would see on Alice's clock at the same moment he saw her turn around, but what time Bob would see on Alice's clock at the actual moment she was turning around, even though the light from this event would take a while to reach him. If so, again, asking what Bob was seeing "at the same moment" Alice was turning around depends on your reference frame. In Alice's frame, this question would be equivalent to "what was Bob seeing on Alice's clock at the moment his clock read 12:55.386?" but in Bob's frame this question would be equivalent to "what was Bob seeing on Alice's clock at the moment his clock read 13:04.998?" And it's not true in either case that Bob would simply see Alice's clock read 20 minutes earlier than 16:00. I can only assume you're basing that on the fact that Alice saw Bob's clock read 12:40 at the moment her clock read 16:00, but as I said in my earlier post to DrGreg, the relativistic Doppler shift equation is not based only on light-speed delays, it's also based on time dilation. In fact, when Bob's clock reads 12:40, he is not at a distance of 20 light-minutes from Alice, in either frame. Since their relative velocity is 0.3846c, then after 40 minutes have passed in his frame, they should be a distance of 40*0.3846 = 15.384 light-minutes apart in Bob's frame. And since she is continuing to move away at 0.3846c, the light will take longer than 15.384 minutes to catch up with her in his frame--you can find the time by solving c*t = 0.3846c*t + 15.384, which gives t = 24.998 minutes for the light to reach her in his own frame. So, the light will reach her 40 + 24.998 = 64.998 minutes after they departed in his frame. But since her clock is only ticking at 1/1.0833 the normal rate in his frame, her clock will only have elapsed 64.998/1.0833 = 60 minutes when the light from this event reaches her.

If you look at it from the point of view of her frame, it's also not true that the light took 20 minutes to reach her. From her point of view, it was Bob's clock that was ticking at 1/1.083333 the normal rate, so when his clock had elapsed 40 minutes, hers had elapsed 40*1.083333 = 43.33332 minutes. Since he was moving away at 0.384615c in her frame, at this moment he would be at a distance of 43.33332*0.384615c = 16.66664 light-minutes away. And since she's at rest in her own frame, the light will take 16.66664 minutes to reach her, so it'll reach her after 43.33332 + 16.66664 = 59.99996 minutes have elapsed on her own clock (it would be exactly 60 if I hadn't rounded off the numbers).

So, in both frames you conclude that the light from Bob's clock reading 12:40 reached Alice when her clock read 16:00, but in neither frame did the light take 20 minutes to travel between Bob and Alice.

 

[Sam Woole]

The turn around occurs when Alice reaches (m), which is at 16:00 according to her own clock. This is something everybody agrees upon. Alice knows it because she is there and looks at her own clock when it happens. Bob does not know that it has happened immediately but has to wait until light has traveled from (m) to (a). When this has happened, Bob can see that Alice has turned round and he can see that her clock was showing 16:00 when it happened..

Now I think I have got it. Here you pointed out that Bob could not apply the 3/2 rate until light from Alice has reached him, that would be 13:30 Bob's time. JesseM also pointed this out to me but I did not take it. I am sorry to both of you. Before now I have insisted that the 3/2 rate be applied as soon as Alice's turnaround took place. I admit I was wrong. Before I accept your second table as correct, I still have one more question. For Bob to see 15:50 (second table) so that he may apply the 2/3 rate, it meant to me that light from Alice after her turnaround has already arrived at Bob, before 16:00. Would you explain why this 15:50 does not mean so?

This is what I said in post #68:
The illusion would be if Alice believed that Bob's clock really was ticking at 2/3 the rate. It is still true that what she sees (after a delay) is Bob's clock's image which ticks at 2/3 the rate of the image of her own clock.

So far I must say thanks again for your patience toward a slow student like me. Here comes my final difficulty. My understanding of your comments about illusion and delay was that, when Alice saw 15 minutes on her clock, Bob also saw 15 minutes on his clock, the same accumulation on both clocks. My understanding must be right because you also gave number 5 as the first delay and double (10) for the second delay, etc. Such a sequence of events means to me, the two clocks will accumulate identical numbers in the end. When the tables showed otherwise (10 minutes more), it was only an illusion created by mathematical works; it was not the true picture. But it was claimed in Relativity to be the true picture, why? It is as if mathematics will remotely affect the workings of mechanical devices and even life forms, as some believe so. This remoteness implies instantaneity. I think it is really hard to believe so. When one twin applies the 3/2 rate many light minutes (or years) away from earth, his brother's clock or body flesh on Earth is instantaneously affected. Is this possible?

 

[JesseM]

Now I think I have got it. Here you pointed out that Bob could not apply the 3/2 rate until light from Alice has reached him, that would be 13:30 Bob's time. JesseM also pointed this out to me but I did not take it. I am sorry to both of you. Before now I have insisted that the 3/2 rate be applied as soon as Alice's turnaround took place. I admit I was wrong. Before I accept your second table as correct, I still have one more question. For Bob to see 15:50 (second table) so that he may apply the 2/3 rate, it meant to me that light from Alice after her turnaround has already arrived at Bob, before 16:00.

Why do you say that? He sees Alice's clock reading 15:50 before he sees her clock read 16:00, which is when she turns around. 15:50 is ten minutes before 16:00, no? And he sees her clock read 15:50 at 13:15 on his clock, which is 75 minutes after she departed at 12:00 according to his clock. 2/3 of 75 is 50, and 15:50 is 50 minutes after she departed at 15:00 according to her own clock.

My understanding of your comments about illusion and delay was that, when Alice saw 15 minutes on her clock, Bob also saw 15 minutes on his clock, the same accumulation on both clocks.

No, DrGreg specifically pointed out that there is no objective way to compare what two clocks read "at the same time" if they are at different locations in space:

In Relativity, the concept of “at the same time” is a relative concept, which is determined by a convention. Using that convention, two different observers can disagree whether two separated events occur “at the same time”.

Once clocks have been separated you cannot apply the time on one clock to events that occur at the position of the other clock.

In Bob's frame, when his own clock has accumulated 15 minutes, Alice's clock has only accumulated 15/1.0833 = 13.847 minutes. Likewise, in Alice's frame during the outbound phase of the trip (note that Alice switches frames when she turns around), when her clock has accumulated 15 minutes, Bob's has only accumulated 13.847 minutes.

But it was claimed in Relativity to be the true picture, why? It is as if mathematics will remotely affect the workings of mechanical devices and even life forms, as some believe so. This remoteness implies instantaneity. I think it is really hard to believe so. When one twin applies the 3/2 rate many light minutes (or years) away from earth, his brother's clock or body flesh on Earth is instantaneously affected. Is this possible?

I like to think of it in geometric terms. Imagine drawing two dots on a piece of paper, and then drawing two paths between them, one a straight line and the other with a bend in it. No matter what coordinate system you use to calculate the lengths of the paths, you will get the same answer for the two lengths, and you will also always find that the straight path is shorter than the non-straight one. Does this mean one path is instantaneously affecting another? No it's just geometry. Similarly, Alice and Bob take two different paths through spacetime between two events (the event of them departing from a single location in space, and the event of them reuiniting at a single location in space), and you can use different space-time coordinate systems to calculate the time elapsed on each path, but all coordinate systems will give the same answer, and you'll always find that the straight path between the points has a shorter amount of time than paths involving changes in direction/speed.

 

[DrGreg]

DrGreg, what was the relative velocity between Alice and Bob in this problem?

Inverting the relativistic Dopper effect equation gives me
$$\frac{v}{c} = \frac {1-R^2}{1+R^2}$$
where $R = 2/3$, so $v = 0.3846c$, or $\gamma = 1.0833$.

Correct.

For the benefit of jtbell and JesseM, I'd like to say that in my presentation to Sam Woole I have tried to avoid quoting any formulas or other results from relativity theory. So I've never invoked any definition of (relative) simultaneity, I haven't discussed time dilation (between frames) or length contraction (between frames), I haven't used any Lorentz transformations or Doppler equations. I wanted to keep the argument as simple as possible and avoid quoting any result that Sam might have difficulty accepting. That's also why I gave an explicit numerical example rather than an algebraic proof.

I believe that in my argument I have used only two assumptions:
- that Alice’s speed relative to Bob is the same as Bob’s speed relative to Alice
- that Alice’s Doppler shift relative to Bob is the same as Bob’s Doppler shift relative to Alice.
Everything else is a logical consequence of those assumptions. (And, in fact, with hindsight, I think I could reword the argument to avoid the first assumption. But that would have complicated the presentation.)


In fact you can deduce the speed from the tables that I quoted in post #103:


12:40 a>>>>>>>>>>>>m 16:00
13:30 a<<<<<<<<<<<<m 16:00

This shows that it takes light 50 minutes to make the round trip from (a) (Bob) to (m) (Alice's furthest position) and back again, so Alice's total distance traveled is 50 light-minutes, according to Bob. And we also know that her total journey takes 130 minutes according to Bob. So her constant speed must be (50 light-minutes) / (130 minutes) = ⁵/₁₃ c, relative to Bob.

And $\gamma$ (which is the same for both halves of the journey) is the total journey time according to Bob divided by the the total journey time according to Alice, 130/120 = 13/12.

 

[DrGreg]

Sam,

I think your questions to me in post #111 have been answered by Jesse in post #112, but if you don't think so, please ask again.

 

[jtbell]

I've never invoked any definition of (relative) simultaneity, I haven't discussed time dilation (between frames) or length contraction (between frames), I haven't used any Lorentz transformations or Doppler equations.

Of course, the assumption that the frequency ratios that Bob and Alice observe are reciprocals (3/2 and 2/3) means that you've implicitly assumed the relativistic Doppler effect. In the non-relativistic Doppler effect, one of them would use the ratio 1 + v/c = 18/13 and the other one, 1 - v/c = 8/13, which are obviously not reciprocals of each other. As their relative speed v decreases (becomes less relativistic), the non-relativistic ratios become closer and closer to the relativistic ones, and more and more nearly reciprocals of each other.

 

[Sam Woole]

Why do you say that? He sees Alice's clock reading 15:50 before he sees her clock read 16:00, which is when she turns around. 15:50 is ten minutes before 16:00, no? And he sees her clock read 15:50 at 13:15 on his clock, which is 75 minutes after she departed at 12:00 according to his clock. 2/3 of 75 is 50, and 15:50 is 50 minutes after she departed at 15:00 according to her own clock.

I am sorry again. I realized my mistake a few minutes after I shut down my computer. You are right.

No, DrGreg specifically pointed out that there is no objective way to compare what two clocks read "at the same time" if they are at different locations in space: In Bob's frame, when his own clock has accumulated 15 minutes, Alice's clock has only accumulated 15/1.0833 = 13.847 minutes. Likewise, in Alice's frame during the outbound phase of the trip (note that Alice switches frames when she turns around), when her clock has accumulated 15 minutes, Bob's has only accumulated 13.847 minutes.
I like to think of it in geometric terms. Imagine drawing two dots on a piece of paper, and then drawing two paths between them, one a straight line and the other with a bend in it. No matter what coordinate system you use to calculate the lengths of the paths, you will get the same answer for the two lengths, and you will also always find that the straight path is shorter than the non-straight one. Does this mean one path is instantaneously affecting another? No it's just geometry. Similarly, Alice and Bob take two different paths through spacetime between two events (the event of them departing from a single location in space, and the event of them reuiniting at a single location in space), and you can use different space-time coordinate systems to calculate the time elapsed on each path, but all coordinate systems will give the same answer, and you'll always find that the straight path between the points has a shorter amount of time than paths involving changes in direction/speed.

On this subject, I think I have not moved over to your point of view. When I mentioned readings on clocks, I was referring to a mechanical device. When you gave numbers such as 15/1.0833 = 13.847, and the geometry, you were referring to math, which I agree totally. But when the math shows 13.847 minutes, will the clock show 13.847 minutes. This was my difficulty, or my inability to believe.

As to whether there is an objective way to compare two clocks, I believe DrGreg's demonstration itself was the objective way. It showed, when Alice saw her own clock to read 15 minutes, she saw 10 minutes on Bob's clock thru the telescope. DrGreg pointed out, there was a 5 minute delay. This is to say, Bob's clock did accumulate 15otherwise Alice could not have seen 10. If Bob looked into his telescope, of course he would find the same number 10 on Alice' clock, from which Bob concludes that Alice's clock has accumulated the same number of minutes as his, 15. To me, this should be the objective way to compare clocks.

Did you see my difficulty here? Your math produced 13.847and 15 in one reference frame, while my mechanical method did 15 for both frames. Will the 13.847 produced in Alice's frame influence the workings of a clock in Bob's frame, remotely?

 

[DrGreg]

Of course, the assumption that the frequency ratios that Bob and Alice observe are reciprocals (3/2 and 2/3) means that you've implicitly assumed the relativistic Doppler effect. In the non-relativistic Doppler effect, one of them would use the ratio 1 + v/c = 18/13 and the other one, 1 - v/c = 8/13, which are obviously not reciprocals of each other. As their relative speed v decreases (becomes less relativistic), the non-relativistic ratios become closer and closer to the relativistic ones, and more and more nearly reciprocals of each other.

For what it's worth, long ago, in my original post https://www.physicsforums.com/showpost.php?p=783491&postcount=68", I gave an argument to show that the blue and red shifts must be reciprocal (assuming that shifts depend only on relative velocity).

Note I've now added another paragraph in post #113.

 

[DrGreg]

Would you explain why this 15:50 does not mean so?

I think JesseM has answered this in post #112.

(By the way, I don’t think JesseM’s post #110 will help you very much. Everything he says is correct but he is quoting some results from Relativity theory that you are probably not familiar with.)

When one twin applies the 3/2 rate many light minutes (or years) away from earth, his brother’s clock or body flesh on Earth is instantaneously affected. Is this possible?

No it isn’t. When Alice starts applying the 3/2 rate, it makes no difference to any time that has already accumulated on her own clock, or to any time that she has already seen accumulate on her image of Bob’s clock. What changes is the relative rate of accumulation after that point. For every extra 10 minutes added to her clock, she sees an extra 15 minutes added to her image of Bob’s clock. This change is due to her change of motion relative to Bob. Nothing has happened to Bob.

You are still thinking in terms of simultaneity, which is confusing you.

Let me give an analogy.

Suppose you and I stand next to each other. You walk 100 metres north. I walk 100 metres northeast. We have both walked 100 metres forward but we are not in the same place.

From your point of view, you are 100m north of the start, but I am only 70.7m north of the start. You have walked further forward (north) than me.

From my point of view, I am 100m northeast of the start, but you are only 70.7m northeast of the start. I have walked further forward (northeast) than you.

Now I turn and walk 100m northwest. You continue to walk 41.4m north, until we both meet again. The point where we meet is 141.4m north of the start.

You have now walked 141.4m (north). I have walked a total of 200m (100m NE + 100m NW). We have walked different distances but are at the same place.

In this analogy, “distance walked forwards” is the analogue of time. We each have our own distance that we have walked. There is no “absolute distance” that can be applied to everybody. 100m N is not the same as 100m NE. If you specify that two places are 100m apart, that does not specify their location – you have to specify a direction as well.

The same goes for time. You cannot say that two events occur 10 minutes apart – you have to specify a “direction in spacetime”, which means you have to specify the motion of the clock that will measure the time. Different clocks (moving at different speeds) will measure different times, in the same way that in the first half of my analogy we disagreed over who had walked “forward” the furthest. (I thought I had walked 29.3m further forward than you. You thought you had walked 29.3m further forward than me.)

In the analogy, the person who changed direction (me) walked further, in total, than the other. Spacetime is slightly different – the person who changes direction (Alice) takes less time, in total, than the other.

 

[DrGreg]

As to whether there is an objective way to compare two clocks, I believe DrGreg's demonstration itself was the objective way. It showed, when Alice saw her own clock to read 15 minutes, she saw 10 minutes on Bob's clock thru the telescope. DrGreg pointed out, there was a 5 minute delay.

This apparent "delay" is a comparison between a time on Alice's clock and another time on Bob's clock. Times on different clocks cannot be directly compared. There is no absolute time, only time relative to an observer.

This is to say, Bob's clock did accumulate 15 otherwise Alice could not have seen 10.

But you cannot say that the moment when Bob's clock has accumulated 15 minutes occurs "at the same time as" the moment when Alice's clock has accumulated 15 minutes. These events cannot be compared absolutely and different observers will disagree which of these events occurred first.

If Bob looked into his telescope, of course he would find the same number 10 on Alice' clock, from which Bob concludes that Alice's clock has accumulated the same number of minutes as his, 15. To me, this should be the objective way to compare clocks.

There is no "objective way" that everyone can agree on. In the same way that, in my analogy in post #118, there is no "objective way" to say who has traveled further forward, because everyone disagrees about which direction is "forward".

 

[JesseM]

For what it's worth, long ago, in my original post https://www.physicsforums.com/showpost.php?p=783491&postcount=68", I gave an argument to show that the blue and red shifts must be reciprocal (assuming that shifts depend only on relative velocity).

Which section of post #68 are you referring to? I may be thinking about this wrong, but it doesn't make sense to me that assuming the shifts depend only on relative velocity is enough to conclude the shifts must be reciprocal...imagine that 2 observers in a Newtonian universe are shooting pellets at each other at a rate of 1 pellet per second, and that both shoot the pellets at the same constant velocity in their rest frame. Wouldn't it be true that the frequency of incoming pellets depends only on the two observers' relative velocity, yet the frequency of incoming pellets when they are moving apart at velocity v is not the reciprocal of the frequency when they are moving towards each other at velocity v?

 

[DrGreg]

Which section of post #68 are you referring to?

The table with Bob, Alice and Ted and the paragraphs immediately before and after.

If Bob and Ted are stationary relative to each other and Alice moves between them, then the Doppler factor for light from Bob to Alice, multiplied by the Doppler factor for light from Alice to Ted must equal the Doppler factor for light from Bob to Ted, which must be 1.

I may be thinking about this wrong, but it doesn't make sense to me that assuming the shifts depend only on relative velocity is enough to conclude the shifts must be reciprocal...imagine that 2 observers in a Newtonian universe are shooting pellets at each other at a rate of 1 pellet per second, and that both shoot the pellets at the same constant velocity in their rest frame. Wouldn't it be true that the frequency of incoming pellets depends only on the two observers' relative velocity, yet the frequency of incoming pellets when they are moving apart at velocity v is not the reciprocal of the frequency when they are moving towards each other at velocity v?

Thanks for an interesting question.

You have made me realize that, in fact, I am making an extra assumption beyond those I explicitly stated. I am assuming that the speed of light does not depend on the speed of the emitter, or to put it another way, that one photon can never overtake another photon traveling in the same direction. (Of course, all of my assumptions are consequences of Einstein’s postulates.)

In your example, the (absolute) speed of the pellet does depend on the (absolute) speed of the emitter.

And in fact my reciprocal argument doesn’t even require “Doppler symmetry” although other parts of my argument do.

In your post #115 when you quote non-relativistic Doppler factors of (1+v/c) and (1-v/c) you are assuming that the speed of light is constant relative to the emitter. If, instead, you were to assume that the speed of light were constant relative to a supposed ether, there would be a different value of c in the two factors (e.g. c = c₀ in the first and c = c₀ + v in the second, which gives reciprocal factors).

 

[JesseM]

Thanks for an interesting question.

You have made me realize that, in fact, I am making an extra assumption beyond those I explicitly stated. I am assuming that the speed of light does not depend on the speed of the emitter, or to put it another way, that one photon can never overtake another photon traveling in the same direction. (Of course, all of my assumptions are consequences of Einstein’s postulates.)
In your example, the (absolute) speed of the pellet does depend on the (absolute) speed of the emitter.

OK, so you're assuming:
1. there is at least one frame where all light-signals travel at c
2. each inertial frame observes the same relationship between the velocity of the emitter and the redshift/blueshift of the signal

This seems like it's probably just a special case of the fundamental postulates of relativity, which can be stated as:
1. there is at least one frame where Maxwell's laws are obeyed, including the fact that all light-signals travel at c
2. each inertial frame observes the same laws of physics

My opinion is that since all the tricky issues of simultaneity and so forth are made explicit in the second version, it's actually better from a pedagogical point of view to discuss the full consequences of Einstein's postulates then to just have it all be implicit as in your doppler shift argument--witness Sam Woole's confusion about what the two clocks will read at the same time, for example, or his misunderstanding that the weird effects of relativity were just optical effects based on light-signal delays.

In your post #115 when you quote non-relativistic Doppler factors of (1+v/c) and (1-v/c) you are assuming that the speed of light is constant relative to the emitter.

That was actually jtbell's post, not mine, but I agree the traditional doppler shift argument assumes the speed of a wave doesn't depend on the speed of the emitter.

If, instead, you were to assume that the speed of light were constant relative to a supposed ether, there would be a different value of c in the two factors (e.g. c = c₀ in the first and c = c₀ + v in the second, which gives reciprocal factors).

What do you mean by "first" and "second" here? The two formulas given by jtbell are just for approaching at v vs. moving apart at v as seen in the frame of the ether, whereas the different values of c you give would be for the ether frame vs. a different frame, right? Or am I misunderstanding?

 

[Sam Woole]

No it isn’t. When Alice starts applying the 3/2 rate, it makes no difference to any time that has already accumulated on her own clock, or to any time that she has already seen accumulate on her image of Bob’s clock. What changes is the relative rate of accumulation after that point. For every extra 10 minutes added to her clock, she sees an extra 15 minutes added to her image of Bob’s clock. This change is due to her change of motion relative to Bob. Nothing has happened to Bob.

DrGreg, these words of yours were describing a picture true to me, though it might not be so to you. You said, every 10 minutes accumulated on Alice's clock, she sees 15minutes added to her image (of Bob's clock). The crucial point here was the word "image." I believe, "added to image" is quite different from "added to clock". Your words "Nothing has happened to Bob" support my belief absolutely. Namely, nothing has added to Bob's clock. This [nothing] means to me, all the images and different rates will not and cannot influence the working of Bob's clock. It follows that when the two clocks met, both will have accumulated the same thing. When you put a different number (10 minutes more) beside Bob's clock, it was nothing but an illusion, created by applying the rates to the images. If we apply colors to photographic images, we certainly are not applying to the original object that produces the images. This picture of mine was deduced from nowhere but your own demonstration. I believe it to be the correct deduction.

You are still thinking in terms of simultaneity, which is confusing you.
Let me give an analogy.
Suppose you and I stand next to each other. You walk 100 metres north. I walk 100 metres northeast. We have both walked 100 metres forward but we are not in the same place. From your point of view, you are 100m north of the start, but I am only 70.7m north of the start. You have walked further forward (north) than me.
From my point of view, I am 100m northeast of the start, but you are only 70.7m northeast of the start. I have walked further forward (northeast) than you.
Now I turn and walk 100m northwest. You continue to walk 41.4m north, until we both meet again. The point where we meet is 141.4m north of the start.
You have now walked 141.4m (north). I have walked a total of 200m (100m NE + 100m NW). We have walked different distances but are at the same place.
In this analogy, “distance walked forwards” is the analogue of time. We each have our own distance that we have walked. There is no “absolute distance” that can be applied to everybody. 100m N is not the same as 100m NE. If you specify that two places are 100m apart, that does not specify their location – you have to specify a direction as well.
The same goes for time. You cannot say that two events occur 10 minutes apart – you have to specify a “direction in spacetime”, which means you have to specify the motion of the clock that will measure the time. Different clocks (moving at different speeds) will measure different times, in the same way that in the first half of my analogy we disagreed over who had walked “forward” the furthest. (I thought I had walked 29.3m further forward than you. You thought you had walked 29.3m further forward than me.)
In the analogy, the person who changed direction (me) walked further, in total, than the other. Spacetime is slightly different – the person who changes direction (Alice) takes less time, in total, than the other.

According to everything I have learned so far, the last sentence above should read: the person who changes direction adds more time to the image of the other guy.

I believe, regardless how much he adds to the image, he does not add anything to the other guy, similar logic as the deduction above.

 

[JesseM]

DrGreg, these words of yours were describing a picture true to me, though it might not be so to you. You said, every 10 minutes accumulated on Alice's clock, she sees 15minutes added to her image (of Bob's clock). The crucial point here was the word "image." I believe, "added to image" is quite different from "added to clock".

Again, Sam, the change in the ticking rate of the image of Bob's clock as seen by Alice has to do with both light-signal delays and genuine time dilation of his clock in her frame. In her frame, his clock is slowed down by a factor of 1.0833, so when 15 minutes pass on his clock, 15*1.0833 = 16.2495 minutes have passed on hers. But since he is rushing towards her, each successive signal has a shorter distance to travel to reach her. He's coming towards her at a velocity of 0.3846c, so in 16.2495 minutes he will have gotten closer to her by 16.2495*0.3846c = 6.2495 light-minutes. So if a given signal takes X minutes to reach her, the next one will only take X-6.2495 minutes to reach her because of that shorter distance. This means that although the actual time between signals coming from Bob in Alice's frame is 16.2495 minutes, the time between her receiving successive signals is 16.2495-6.2495 = 10 minutes.

Notice that if there was no time dilation, so that his clock really did send a signal every 15 minutes in her frame, then she would receive signals every 15-6.2495 = 8.7505 minutes, not every 10 minutes. So again, you have to take both light-signal delays and time dilation into account to understand why she sees these signals every 10 minutes.

edit: sorry, minor mistake in that last paragraph--if his clock sent a signal every 15 minutes in her frame, then he would get closer by 15*0.3846c = 5.769 light-minutes between signals rather than 6.2495 light-minutes, so Alice would receive signals every 15-5.769 = 9.231 minutes.

 

[Janus]

DrGreg, these words of yours were describing a picture true to me, though it might not be so to you. You said, every 10 minutes accumulated on Alice's clock, she sees 15minutes added to her image (of Bob's clock). The crucial point here was the word "image." I believe, "added to image" is quite different from "added to clock". Your words "Nothing has happened to Bob" support my belief absolutely. Namely, nothing has added to Bob's clock. This [nothing] means to me, all the images and different rates will not and cannot influence the working of Bob's clock. It follows that when the two clocks met, both will have accumulated the same thing. When you put a different number (10 minutes more) beside Bob's clock, it was nothing but an illusion, created by applying the rates to the images. If we apply colors to photographic images, we certainly are not applying to the original object that produces the images. This picture of mine was deduced from nowhere but your own demonstration. I believe it to be the correct deduction.
According to everything I have learned so far, the last sentence above should read: the person who changes direction adds more time to the image of the other guy.
I believe, regardless how much he adds to the image, he does not add anything to the other guy, similar logic as the deduction above.

Your deduction is not correct. In this example, we have Alice traveling out and returning to Bob. IOW, at the end we have Bob and Alice's clocks sitting right next to each other again. During the entire duration of the experiment Bob and Alice have watched each other's clocks. And during that time Alice sees Bob's clock accumulate more time than her's and Bob sees Alice Clock acumulate less than his, ending with Bob and Alice standing next to each other and looking at both of their clocks sitting next to each other.

So is it your contention that when Bob and Alice are sitting there at the end, looking at their two clocks in the same room as they are, the images of the time reading on the clocks that reach their eyes are not what the clocks actually read?

If not, then at what point do the images magically shift from illusion to reality?

 

[DrGreg]

My opinion is that since all the tricky issues of simultaneity and so forth are made explicit in the second version, it's actually better from a pedagogical point of view to discuss the full consequences of Einstein's postulates then to just have it all be implicit as in your doppler shift argument--witness Sam Woole's confusion about what the two clocks will read at the same time, for example, or his misunderstanding that the weird effects of relativity were just optical effects based on light-signal delays.

Well, maybe you are right, who knows? I thought I’d try a different approach, where instead of actually defining relative simultaneity, I would leave it undefined and effectively take the view that once two events are separated by a distance, then it is impossible (meaningless) to say whether they are simultaneous or not. Perhaps I should have emphasised this at the beginning of my original message. It would seem that my approach hasn’t succeeded.

(The question of time dilation (between frames) is linked to the definition of simultaneity i.e. the method of synchronising clocks within a frame. The standard Einsteinian convention as implied by his postulates is the one that makes most sense, and imposes an orthogonal co-ordinate system on spacetime, but nevertheless other conventions are possible, as I have debated in other threads: https://www.physicsforums.com/showthread.php?t=88636". I don’t intend to resurrect those debates in this thread.)

What do you mean by "first" and "second" here?

(1+v/c) and (1-v/c) in that order

The two formulas given by jtbell are just for approaching at v vs. moving apart at v as seen in the frame of the ether, whereas the different values of c you give would be for the ether frame vs. a different frame, right? Or am I misunderstanding?

I was interpreting the formulas to refer to A’s signals relative to B and B’s signals relative to A, respectively, which is why A and B would disagree over the speed of light relative to themselves on the assumption of an ether. (Sorry for not noticing who posted what, by the way.)

 

[Sam Woole]

Your deduction is not correct. In this example, we have Alice traveling out and returning to Bob. IOW, at the end we have Bob and Alice's clocks sitting right next to each other again. During the entire duration of the experiment Bob and Alice have watched each other's clocks. And during that time Alice sees Bob's clock accumulate more time than her's and Bob sees Alice Clock acumulate less than his, ending with Bob and Alice standing next to each other and looking at both of their clocks sitting next to each other.
So is it your contention that when Bob and Alice are sitting there at the end, looking at their two clocks in the same room as they are, the images of the time reading on the clocks that reach their eyes are not what the clocks actually read?
If not, then at what point do the images magically shift from illusion to reality?

Janus, you are right in that I was contending that the numbers or rates are illusions in the observers' mind, or eyes if you like. Though I did and still do accept JesseM and DrGreg's numbers, yet I do not believe Bob's clock will show these numbers. Let me explain my belief and see whether you can correct me. The following is part of DrGreg's first demonstration but one more element was added by me.

Both clocks of Bob and Alice read 00 minutes at start.
After 15 minutes of motion:

Bob's clock-----Image ------Alice own clock.
----?--------------10---------------15---------

Here Alice found (from the image she saw) that Bob's clock was accumulating at a lower rate, 2/3. By applying this rate to Bob's clock, I believe Bob's clock will not budge. I put a question mark under Bob so you may correct me if you think Bob's will obey.

Alice continued moving and returning and applying rates. I believe Bob's clock will do its own work; will not be affected. What was affected were the numbers in Alice's mind.

I also brought up the instantaneity problem. When Alice was applying those rates, she was a long way from Bob. If Bob's clock will be affected, it means the affect was taking place instantaneously.

 

[russ_watters]

The bigger question (asked, but not yet answered) is: when Bob's and Alice's clocks are sitting next to each other at the end of the experiment, do they read the same time? You do understand that this experiment has actually been performed, right? (someone mentioned it already)

My personal opinion of threads like this is that they really should start with the acceptance of the end result of the experiment. The past week of explaining the math and the nitty-gritty of the experiment won't help if at the end, you don't accept the real results of the experiment. Also, starting with the realization of a result that you didn't expect puts you into a 'looking for my mistake' mode instead of a 'trying to justify my preconception' mode, making the learning process go much more smoothly.

Sometimes it is hard to grasp the difference (or lack, thereof) between perception and reality when reading a clock through a telescope. Accepting up front that perception is reality - but that reality depends on your frame of reference - would help a lot in understanding this issue.

 

[DrGreg]

DrGreg, these words of yours were describing a picture true to me, though it might not be so to you. You said, every 10 minutes accumulated on Alice's clock, she sees 15 minutes added to her image (of Bob's clock). The crucial point here was the word "image." I believe, "added to image" is quite different from "added to clock".

The image shows something that has already happened to Bob and his clock some time ago in the past. It takes time for the image to travel from Bob to Alice. There is nothing false about the image itself; the only problem is deciding when the image was created. This is the real problem. You are using your own notion about how to calculate when something happened and the way you are doing it isn’t compatible with Relativity, I’m afraid.

In space, if you and I stand next to each other, then you walk 15 metres and I walk 10 metres, you cannot conclude that we are now 5 metres apart; it depends which direction we walked in.

You may not believe it, but nevertheless it is true that time behaves the same way. If you and I stand next to each other, then you walk for 15 minutes (by your clock) and then shout, and meanwhile I walk for 10 minutes (by my clock) and then shout, you cannot conclude our shouts occurred 5 minutes apart; it depends what speeds we traveled at. This might sound ludicrous, but it is true. Your sense of time is based on your experience of moving at speeds that are tiny compared with the speed of light, so these strange effects are not noticeable to us human beings.

If time could be added up or subtracted the way you think it can, then the speed of light would not be constant, high-speed muon particles would not decay much more slowly than low-speed muons, and “sat nav” car navigation systems would not work the way they do.

According to everything I have learned so far, the last sentence above should read: the person who changes direction adds more time to the image of the other guy.

I believe, regardless how much he adds to the image, he does not add anything to the other guy, similar logic as the deduction above.

Nobody is actually adding anything to anything. They are just seeing what is there.


Please also read what Janus said in post #125, which I agree with. See also what russ_watters said in post #128. I think he makes a good point.

________________

I am sorry that I will not be able to answer any more questions during October, as I will be on vacation. I hope to be back here by 1 November. Goodbye for now.

 

[jackle]

I believe, regardless how much he adds to the image, he does not add anything to the other guy...

Hello, sorry for coming in late.

It may help to understand that according to relativity, because nothing can travel faster than light, different points in space that are very far apart are effectively separated by time as well as space. Even light can not connect the two regions in a short time, so they experience a degree of isolation that is a part of the laws of physics.

Once you truly grasp this, you can let go of "what is really happening to the other guy right now". The images or even slower signals are all you will ever get. The other guy is separated in time and space with their own time-system. Therefore, you are correct in that the clocks in no way have an affect on each other, but they have traveled differently through space-time. (It isn't a trick of the mind either).

The only other way I know to compare clocks is to travel the gap and compare clocks. I believe we have discussed both these two methods now (sending images and crossing the distance).

Finally, relativity has been supremely verified scientifically. Time dilation is absolutely real and has nothing to do with the mechanics of clocks, and of course the maths does work out! You don't have to believe it, but you should understand that it has been tested enormously and it is a brilliant theory that is well worth learning properly and taking very seriously.

Good luck

 

[Sam Woole]

In the last few days I went back to my former posts and realized that I had more mistakes than I admitted. I truly believe there is something preventing me from learning a high science such as the Relativity. I am going to do more reading and try harder.

In the meantime I still wish to say, the Relativity itself could be blamed for my inability as well as other people's. The time dilation was one big culprit. All the books I tried I had relative uniform motion causing time dilation. Atomic clock inventor Dr. Louis Essen understood it to be relative uniform motion, and a lot of others whose writings had come to light. Now there came first acceleration, and then Doppler effects.

I thank everybody again and wish to see you on other new topics.

 

[JesseM]

In the meantime I still wish to say, the Relativity itself could be blamed for my inability as well as other people's. The time dilation was one big culprit. All the books I tried I had relative uniform motion causing time dilation. Atomic clock inventor Dr. Louis Essen understood it to be relative uniform motion, and a lot of others whose writings had come to light. Now there came first acceleration, and then Doppler effects.

It's still only relative velocity that causes time dilation in any particular frame--the rate a clock moving at velocity v ticks is always $$\sqrt{1 - v^2/c^2}$$. But for two clocks two be compared at different times in the same location, they have to move apart and then one has to turn around (accelerate), and if you sum up $$\sqrt{1 - v^2/c^2}*dt$$ for a bunch of small time intervals dt along each clock's path to get the total time elapsed on that clock, the math works out so it's always the one that accelerated that shows less total time. So its not a case of two competing notions of what causes time-dilation, it's part of a single self-consistent theory (as for doppler shift, that's only for what each observer will see using delayed light-signals, and the doppler shift equation can be derived from the time dilation equation + the assumption that light always moves at c).

 

[eon_rider]

Great thread. First Post

I wish I had found this thread earlier and I'm still re-reading it. It prompted me to join the forum. It’s great. Thanks to all who asked and answered questions and patiently explained. The more posts I’m reading the more I’m hoping to “get it.” LOL.

Anyway.

Quick question on time dilation and ageing:
Has there been an experiment done to see if when atomic clocks run slower on the space shuttle or satellites due to S.R. does this necessarily affect biological ageing processes?

Is it assumed that if an cesium atom oscillates slower,
then atoms oscillating slower in the human body
must cause a person to age slower?
Is it possible that it doesn't cause a person to age slower?

All the best,

Eon.

PS. I guess what I’m getting at is this would solve the twin ageing paradox without needing to change the way clocks slow down at very high speeds

 

[russ_watters]

I wish I had found this thread earlier and I'm still re-reading it. It prompted me to join the forum. It’s great. Thanks to all who asked and answered questions and patiently explained. The more posts I’m reading the more I’m hoping to “get it.” LOL.

Welcome aboard.

Quick question on time dilation and ageing:
Has there been an experiment done to see if when atomic clocks run slower on the space shuttle or satellites due to S.R...

Yes! (to both, and there have been many other experiments). GPS satellites are the best practical example. There are 24 (plus several spares) orbiting the Earth at various inclinations, all carrying multiple atomic clocks with single-digit nanosecond accuracies. The combined effects of special and general relativity causes these clocks to gain 38 microseconds per day. The clocks are pre-programmed prior to launch to tick slower by 38 microseconds per day, and as a result, they stay in sync with clocks on the ground. They are checked every time they pass over a ground-station.

More here: http://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

...does this necessarily affect biological ageing processes?

Unfortunatly, there are no biological processes that occur fast enough to actually observe relativistic effects in them, but there is no reason to think that they are not similarly affected. Time dilation is currently observed in many different types of processes (not just in clocks), so there is good evidence that it is time itself that is affected, not just the tick rate of clocks. Also, atomic clocks don't work via mechanical processes (it isn't really an oscillation of the atom) and thus aren't affected by things like friction or gravitational force (ie, a pendulum clock would be affected by gravity, even without time dilation).

Is it assumed that if an cesium atom oscillates slower,
then atoms oscillating slower in the human body
must cause a person to age slower?

Yes.

Is it possible that it doesn't cause a person to age slower?

I guess you can't entirely rule out anything, but it would require an awful lot of current physics to be spectacularly wrong.

I guess what I’m getting at is this would solve the twin ageing paradox without needing to change the way clocks slow down at very high speeds

Well, the paradox is just an analytical device. In this thread, they aren't even using twins, they are just comparing clocks because it is more straightforward that way. The issue works the same either way, though.

 

[jackle]

Biological clocks and passage through time

...Unfortunatly, there are no biological processes that occur slow enough to actually observe relativistic effects in them...

I thought hard about this and I think you meant to say there are no biological processes quick enough to use, because we need something that repeats every billionth of a second or something don't we? Biological processes are generally pretty slow aren't they?

Leading on from this, I imagined a colony of say 10 billion bacteria, where one reproduces every billionth of a second. If you count the bacteria, you might be able to work out the time. Could this be used to verify biological time dilation? I guess this would be far too inaccurate!

...Time dilation is currently observed in many different types of processes (not just in clocks), so there is good evidence that it is time itself that is affected..

I know what you mean by time being affected, but really I think it is our passage through time that is affected. (I agree that it is the nature of time itself that underlies time dilation, not the type of clocks used).

 

[russ_watters]

I thought hard about this and I think you meant to say there are no biological processes quick enough to use, because we need something that repeats every billionth of a second or something don't we?

Yep, nice catch (fixed now).

Biological processes are generally pretty slow aren't they?

Yes.

Leading on from this, I imagined a colony of say 10 billion bacteria, where one reproduces every billionth of a second. If you count the bacteria, you might be able to work out the time. Could this be used to verify biological time dilation? I guess this would be far too inaccurate!

Well, just like with a half-life, if you watch enough bacteria, you can get a reasonably accurate picture of how fast they reproduce. The trick is making the starting conditions identical (and being able to watch 10 billion at once). But I think it could work.

I know what you mean by time being affected, but really I think it is our passage through time that is affected. (I agree that it is the nature of time itself that underlies time dilation, not the type of clocks used).

Yes, maybe that could have been worded better.

 

[JesseM]

Another point is that the laws of chemistry which underlie biological processes are all thought to be based on electromagnetic interactions between molecules, and the equations of electromagnetism have the mathematical property of "Lorentz-invariance", which means they must work the same way in the different inertial reference frames used in SR, which means the speed of all electromagnetism-based clocks moving at velocity v in a given frame must be observed to slow down by $$\sqrt{1 - v^2/c^2}$$. The only way biological processes wouldn't slow down is if they did not obey the known laws of electromagnetism.

To understand what Lorentz-invariance means mathematically, it might be a little easier to first look at the meaning of Galilei-invariance. In Newtonian physics, if you have two coordinate systems moving at velocity v relative to each other along their respective x-axes, and one uses coordinates (x,y,z,t) while the other uses coordinates (x',y',z',t'), then to transfrom between the two coordinate systems you'd use the "Galilei transform" here:

$$x' = x - vt$$
$$y' = y$$
$$z' = z$$
$$t' = t$$

and
$$x = x' + vt'$$
$$y = y'$$
$$z = z'$$
$$t = t'$$

To say a certain physical equation is "Galilei-invariant" just means the form of the equation is unchanged if you make these substitutions. For example, suppose at time t you have a mass $$m_1$$ at position $$(x_1 , y_1 , z_1)$$ and another mass $$m_2$$ at position $$(x_2 , y_2 , z_2 )$$ in your reference frame. Then the Newtonian equation for the gravitational force between them would be:

$$F = \frac{G m_1 m_2}{(x_1 - x_2 )^2 + (y_1 - y_2 )^2 + (z_1 - z_2 )^2}$$

Now, suppose we want to transform into a new coordinate system moving at velocity v along the x-axis of the first one. In this coordinate system, at time t' the mass $$m_1$$ has coordinates $$(x'_1 , y'_1 , z'_1)$$ and the mass $$m_2$$ has coordinates $$(x'_2 , y'_2 , z'_2 )$$. Using the Galilei transformation, we can figure how the force would look in this new coordinate system, by substituting in $$x_1 = x'_1 + v t'$$, $$x_2 = x'_2 + v t'$$, $$y_1 = y'_1$$, $$y_2 = y'_2$$, and so forth. With these substitutions, the above equation becomes:

$$F = \frac{G m_1 m_2 }{(x'_1 + vt' - (x'_2 + vt'))^2 + (y'_1 - y'_2 )^2 + (z'_1 - z'_2 )^2}$$

and you can see that this simplifies to:

$$F = \frac{G m_1 m_2 }{(x'_1 - x'_2 )^2 + (y'_1 - y'_2 )^2 + (z'_1 - z'_2 )^2}$$

In other words, the equation has exactly the same form in both coordinate systems. This is what it means to be "Galilei invariant". More generally, if you have any physical equation which computes some quantity (say, force) as a function of various space and time coordinates, like $$f(x,y,z,t)$$ [of course it may have more than one of each coordinate, like the $$x_1$$ and $$x_2$$ above, and it may be a function of additional variables as well, like $$m_1$$ and $$m_2$$ above] then for this equation to be "Galilei invariant", it must satisfy:
$$f(x'+vt',y',z',t') = f(x',y',z',t')$$

In relativity, instead of using the Galilei transformation to transform between coordinate systems moving at velocity v relative to each other, you use the Lorentz transformation:

$$x' = \gamma (x - vt)$$
$$y' = y$$
$$z' = z$$
$$t' = \gamma (t - vx/c^2)$$
where $$\gamma = 1/\sqrt{1 - v^2/c^2}$$

and

$$x = \gamma (x' + vt')$$
$$y = y'$$
$$z = z'$$
$$t = \gamma (t' + vx'/c^2)$$

So just as with Galilei-invariance, Lorentz-invariance means that if you take some equation for a law of physics written in terms of x,y,z,t coordinates and use the Lorentz transform to make substitutions and rewrite this equation in terms of x',y',z',t' coordinates, the equation will end up looking the same as if you had just replaced x with x', y with y', z with z' and t with t'--the equation should have exactly the same form in both coordinate systems. This means that any equation that's Lorentz-invariant should satisfy:
$$f( \gamma (x' + vt' ), y' , z', \gamma (t' + vx' /c^2 ) ) = f(x' ,y' ,z' , t')$$

As long as all the fundamental equations of physics are Lorentz-invariant, it must be true that you'll see phenomena like time dilation and Lorentz contraction...perhaps this will help answer the question asked by Sam Woole and Sherlock about what the "physical explanation" for these phenomena is. If you were writing a computer program to simulate an imaginary world with any laws of physics you wanted, and all the equations you picked to govern the simulation happened to have this mathematical property of Lorentz-invariance, then you would automatically see time dilation and Lorentz contraction in the simulated world, whether you had planned it that way or not.

 

[eon_rider]

thank you.

The advanced maths went over my head.

Thanks for the welcome.
And the great explanations from all, as to how all the biological processes known to us so far, go too slowly to measure against time dilation. It would be cool to devise an experiment that could be done on the space shuttle on time dilation and ageing but I guess it’s just impossible at the moment.

Perhaps one day we can build faster probes or rockets that can pick up either “changes or non-changes” in bacteria growth times, or some other human biological process.

It’s all good.

Many thanks.

Eon.

 

[jtbell]

One way to state the fundamental principle of relativity is that if you're inside a box moving at constant velocity relative to an inertial reference frame, there is no way that you can find out how fast you're moving, or even whether you're moving at all, using experiments that are contained inside the box. You have to look outside and observe your motion relative to external landmarks.

If non-biological clocks experience time dilation but biological clocks don't, then you could figure out your state of motion while riding inside a sealed box, by comparing the behavior of the two different kinds of clocks. This would violate the principle of relativity.

The principle of relativity leads to many other conclusions which have been tested thoroughly, so we have strong reason to believe that it applies here as well, in the absence of experimental evidence to the contrary.

Nevertheless, it would be very useful and important to test whether biological clocks really do undergo time dilation, when it becomes feasible. If it turns out that they don't, someone will probably get a Nobel Prize for it!

 

[Sam Woole]

It's still only relative velocity that causes time dilation in any particular frame--the rate a clock moving at velocity v ticks is always $$\sqrt{1 - v^2/c^2}$$. But for two clocks two be compared at different times in the same location, they have to move apart and then one has to turn around (accelerate), and if you sum up $$\sqrt{1 - v^2/c^2}*dt$$ for a bunch of small time intervals dt along each clock's path to get the total time elapsed on that clock, the math works out so it's always the one that accelerated that shows less total time. So its not a case of two competing notions of what causes time-dilation, it's part of a single self-consistent theory (as for doppler shift, that's only for what each observer will see using delayed light-signals, and the doppler shift equation can be derived from the time dilation equation + the assumption that light always moves at c).

JesseM, there is no problem for me to accept that "the math works out so it's always the one that accelerated that shows less total time." My point here was how do we determine who accelerated. In DrGreg's demonstration, a third observer Ted created the impression that Alice was the only one that accelerated. Bob never. But it is my understanding that Bob could also be considered as the one that accelerated.

According such understanding of mine, from Bob's viewpoint Alice moves away and returns to him; from Alice's viewpoint Bob moves away and returns to her. According to this understanding of mine, either observer will find the other has accumulated more time (10 minuses), a symmetrical result.

DrGreg's demonstration was designed in such a way that any observer will travel 2 hours to finish the trip. When mathematical work produced 10 more minutes, it was apparently a contradiction to the design. In my opinion
the only sensible and logical action to take is to look into the contradiction, rather than looking into the clock.

I believe this was exactly why it is so hard to learn relativity. Einstein and his supporters shifted from one idea to another. When relativists produced a longer time interval by means of math, they blamed clocks, shifting from math to clocks. I do not think it is good science.
Accountability is the word. I felt you were dealing with the wrong suspect.

I do not know whether relativistic math right or wrong. But another guy named Mark McCutcheon claimed Einstein's math had many fatal errors. You can
read about his claims here:
[Well-known crackpot link deleted]
where you will find he also understood relative motion in the same fashion as I did.