Delta1212

I'm trying to figure out why what I did disagrees with what you did. I'm pretty sure that it's a result of mixing up proper time with coordinate time, but I'm looking to determine exactly where the error is. If everyone agrees on the time that A believes is simultaneous with B meeting C, even if they all disagree that it is actually simultaneous, then you could get a measurement of the proper time that A measures between B leaving and the moment on A's clock that A believes is simultaneous with B's passing of C, but this would not be equal to the proper time on B's clock for the elapsed time of the journey. Right?

Ok, I have another question then. Would every frame agree upon the time at which A believes B to have arrived at C according to A's frame. In other words, if A sets an alarm to go off at the time on A's clock that A believes is simultaneous with B reaching C, would observers in every frame agree on the time that A's clock read when the alarm went off, but disagree that it was simultaneous with B reaching C? I'm hoping that was clear enough to understand.

Ok, so A cannot measure a proper time between B leaving and B reaching C, because A is not present when B reaches C. A's clock will read one time (edited to add: in frame A) as being simultaneous with B reaching C, but an observer in a different frame will read a different time on A's clock as being simultaneous with B reaching C. Therefore the time that elapses on A's clock between B leaving and reaching C is not frame invariant and is not proper time, but coordinate time. Did I get that right?

I am currently walking toward a table that is 10 feet away at a speed of 10 feet per minute. Exactly one year ago, I must have been 1,000 miles away from the table.

xyzt: Can I ask some questions about the equations in the post where you did all the math? It's possible I'm misinterpreting how you're using some of the variables and/or not understanding how things are being plugged into the formulae. If that's the case, it would explain why I don't think your math adds up, because then I'm equating values that aren't meant to be the same. You don't have to actually do any math, I just want to know where a few things came from.

I pointed out something that seems like a mistake to me. It is possible that I am wrong, in which case, I need to understand why. If I see something contradictory in the explanation, I will point it out because it means that either there is a mistake in the explanation or I haven't understood it properly. Either way, there needs to be a correction for us to be on the same page. I apologize if you don't like me trying to poke holes in what you're saying, and I do understand that, assuming you are totally correct, this can be frustrating. If I don't try to poke holes, though, I'm not going to figure out how it works. I don't mean to offend. You suggested I learn to follow the math, and I'm making a genuine effort to do that. I believe I understand what proper time and coordinate time are, but if I do, then you made a mistake, so if you're correct, then I'm obviously misinterpreting something. I asked for specific numbers because if I can compare actual examples of proper time, I'll have a better grasp on where my understanding appears to differ from yours so that it can be corrected. It's entirely possible that the fault lies with me, but I'm having trouble figuring out exactly where I'm going wrong, and simply saying that proper time is frame invariant doesn't seem to be helping because I thought I already understood that.

I don't keep giving you negatives. I rarely give people anything, actually.

Wouldn't it be equivalent to accelerating from [math]v_B[/math] to [math]v_C[/math] since the first time measurement is made at [math]v_B[/math] and the second at [math]v_C[/math]? Or is that just semantics? Honest question. For the sake of comparison, what would [math]\tau''_C[/math] be if C did not reset it's clock on passing B?

A bit like crushing up all the ice from a berg, evenly distributing it over the course of the Titanic's route and seeing whether it creates enough drag to stop it?

Alright, just in case I'm actually getting confused, because I do like to double check myself to make sure I'm not screwing up. xyzt: Going off of your muon example. If the distance is measured to be 2 ly and, for the sake of simplicity the velocity is measured to be 13/15c (both from Earth's frame of reference), can you tell me: The time the muon takes to travel that path in its own frame, the time that the Earth measures the muon to take in Earth's frame, the proper time of the muon, and the proper time of Earth, and explain how to find each? I realize some of those values will be the same, I just have a feeling this will be easier to understand if I can compare some actual numbers and see what they represent. Edit: Like I said, I'm new to this, so if I'm making a fundamental mistake somewhere, I really need to figure out where any confusion of mine rests, and that will be easier if I can see the math done correctly than if I keep attempting it the way I've understood it which, if wrong, won't help me any.

The fact that it should be d1 and d2 is actually the entire basis of my argument. If you go back and look at xyzt's post that I'm taking this from, d and v aren't generic variables for distance and velocity. He specifically defines them as d being the distance travelled by C as measured by A and v as the velocity of C as measured by A. He then does tau_A = d/v *sqrt(1-(v/c)^2) from frame C. d here should be d(2). However, he takes that expression and subtracts it from d/(u+v)(sqrt(1-(u/c)^2-sqrt(1-(v/c)^2)+d/v*sqrt(1-(v/c)^2) which defines the proper time experienced by B on the first leg with C on the second leg, where all of the d's are d(1) as if d(1) and d(2) are the same variable. He then uses the result to explain why the whole thing falls apart, when he was actually using a single variable to define to separate distances, which resulted in him subtracting the proper time experienced by C over the course of the experiment while thinking he was subtracting the proper time of A over the course of the experiment. When I used the same variables that he did, I derived that contradiction because it's wrong. Please go back and check his math to see whether I'm way off base here. It's in post #73. I wasn't trying to do that. I was pointing out that you did do that.

Proper time is the time read by a clock between two events that the clock is present at. Everyone agrees on what a clock measures in its own frame. This is why proper time is frame invariant. If the Earth measures a muon traveling to Earth from distance d (as measured by Earth) at velocity v (as measured by the Earth), then d/v * sqrt(1-(v/c)^2) will give you the proper time of the muon as it travels that path in the frame of the muon. But d/v will be the time that the Earth measures the muon taking in Earth's frame. It will, however, agree that the muon experiences d/v *sqrt(1-(v/c)^2) time during that period because that is the muon's proper time, which is agreed upon in all frames. Proper time being frame invariant does not mean that every frame measures the same time between two events. It just means they all agree on what each other measures.

If that is the time the muon takes as measured by the Earth, what would the proper time of the muon be?

Let me take this one step at a time and see if I can find a fault in my understanding. A measures C traveling a distance d at velocity v. A measures the time that this takes as being d/v. Is this, at least, correct?

Ok, right. You were attempting to find the proper time of A using the frame of C. To do that, you multiply the time measured in frame C by sqrt(1-(v/c)^2). Since you had the distance d and the velocity v (since C measures A traveling towards it at the same speed that A measures C traveling at). To find time you do d/v. And you wind up with the equation: tau_A = d/v * sqrt(1-(v/c)^2) or (d *sqrt(1-(v/c)^2))/v, as you said. Awesome. Except that in frame C, d does not equal the distance traveled by A as measured by C. It equals the distance traveled by C as measured by A. You found the time measured in frame C by dividing the distance measured in frame A by the velocity of A as measured in frame C. Here, let's try this out. In your example, in frame A, A starts counting from the moment B leaves, which A measures as being simultaneous with the moment that C is d distance away from A, traveling towards it at velocity v. A therefore measures the total time from the beginning of the experiment (which in frame A is also when C started moving toward A along a path of length d) as being d/v. Now, you say that frame C measures A's proper time as being d/v * sqrt(1-(v/c)^2). Since proper time is agreed upon in all frames, then A's proper time as measured in frame A must be equal to A's proper time as measured in frame C. A measures it's proper time as d/v. C measures A's proper time (according to your math) as d/v *sqrt(1-(v/c)^2). Therefore: d/v = d/v * sqrt(1-(v/c)^2) 1 = sqrt(1-(v/c)^2) 1 = 1-(v/c)^2 1 + (v/c)^2 = 1 (v/c)^2 = 0 v/c = 0 v = 0 So if d/v = d/v * sqrt(1-(v/c)^2), then for any value v, v = 0. If v = 1, then v = 0. 1 does not equal 0. Clearly something is wrong. Either: 1. I can't do algebra, in which case please point out the mistake. 2. The proper time that A measures for frame A between the time it measures C traveling along distance d at velocity v and the time it measures C as finishing traveling along distance d at velocity v cannot be expressed as d/v and I am wrong. (In which case, please explain because I clearly have a misconception about something here). 3. The proper time of A as calculated from frame C is not d/v * sqrt(1-(v/c)^2), in which case the conclusions you draw from that are wrong.

Please show me where you did that. I honestly can't find it.

Please re-read your own post: "On the other hand, the proper time accumulated by A can be calculated by a frame comoving with C, F_C as : tau_A=d/v*sqrt(1-(v/c)^2) (all frames agree on proper time so we might as well use F_C)" Where d is the length as measured in frame A and v is A's velocity as measured in frame C.

Actually, you said that B and C begin moving simultaneously from A's frame along a segment measuring length d (also from A's frame). You then measured the proper time that A would measure if it traveled d distance at C's speed (in other words, A's proper time as measured from C's frame). Great, everyone agrees on proper time. Except that in frame C, C did not reach distance d from A and B simultaneous to the moment that they separated, because C is in a different location and moving at a different velocity to A. In C's frame, it is not d distance away at the start of the experiment, so while it does measure A approaching it at the same velocity that A measures C approaching, they do not agree on the time that the experiment started, and therefore on the distance that needs to be traveled. You use d/v as the time that C measures A to take while finding A's proper time, but d is the length of the distance traveled since the start of the experiment as measured in frame A, not in frame C.

2. This doesn't make sense, especially- Oh, I just realized why xyzt thinks setting B and C to the same speed makes B irrelevant. Ok, here's why I disagree with both of these points anyway: This isn't an example of C measuring a discrepancy in elapsed proper time since the beginning of the experiment with A in the way that the twin paradox is generally formulated as a discrepancy between the time experienced by one twin at rest with another twin that takes a non inertial path. Rather, it is a method of measuring the time experienced along that path without sending someone the full length of the trip. Here, say we have a pair of twins all set to do the experiment, but they can't agree on which is to stay home and which is to leave. They are about to give up in despair when a friend from NASA comes to them and says "I have a probe returning from deep space to Earth that will be passing through the area you would have turned your ship around. If we send a clock out to meet it, we can record the length of time the clock measures over the course of the first leg of the trip that you would have taken. Then we can use the probe's internal clock to measure the time experienced on the second leg of the trip." "Nothing will be returning having aged less than those who stated behind, but we can at least experimentally verify the length of time the traveling twin would have experienced so you can compare it with the amount both of you actually age while staying here on Earth." C doesn't say "hey, I'm right not you" upon meeting B, because C isn't accepting B's time as the correct time at that moment, it is accepting that as the proper time of the first leg of a hypothetical twin experiment. C didn't start at the same location as the at rest partner, nor travel between that partner and the "turnaround point" where B passes C. C cannot, therefore, measure a proper time between those two events on its own clock, but it can see the proper time that B measured on that leg (which B actually traveled) and say "Ah, now I know how much time would be experienced by the twin on his trip away. As I am about to travel the path such a twin would have taken on the way back, I can add B's time to the time I experience between this moment and the one when I pass A and determine how much time such a twin would have experienced had he traveled away from A (with the velocity of B) to the point at which BC took place, and then turned around and traveled back to A (with the velocity of C)." No, it's good. I did already understand most of that, but it's nice to have confirmation that I'm not misinterpreting what proper time is. I already basically grasped the concept of it (that everyone would agree on the time a particular clock measured between events it was present at), but I didn't know it was called that or exactly how to calculate it from measurements taken in another frame until yesterday.

Ok, here's a point I need clarification on. As far as I am aware, proper time is the time recorded by an observer between two events, both of which the observer was at. You can determine the proper time measured by A between AB and AC. You can measure the proper time recorded by B between AB and BC. You can measure the proper time recorded by C between BC and AC. You cannot record the proper time measured by A between AB and BC because A was not at BC. If this is the case, then A cannot measure proper times between AB and BC or between BC and AC, meaning that the measurements A makes of the times the other clocks take to travel between these events are coordinate times and a gamma correction must be applied. I really do need someone to check what I just said.

B is moving with respect to A. They should not measure the same time between AB and BC. Edit to respond to your edit: Hold on. Let me go back and look at what I did.

Let me try doing some math for that then. Let's use Clock A's frame of reference. Let's say that A observes Clock B traveling at 0.8c and Clock C traveling at 0.5c. Let's also say that B passes C at a point 10 ly from A. A will measure B traveling between events AB and BC over the course of 12.5 years. A will also measure C traveling from between BC and AC over the course of 20 years, so A's elapsed proper time between AB and AC will be 32.5 years. B's proper time between AB and BC is 12.5 * sqrt(1-(0.8/c)^2) = 7.5 years C sets it's time to 7.5 years at BC and then measures a further proper time between BC and AC of 20 * sqrt(1-(0.5/c)^2) = 17.32 years 7.5 + 17.32 = 24.82 years 24.82 < 32.5 Which agrees with your assertion that, if u < v (which it is here), tau_C will be less than tau_A. However, if I reverse the speeds, but keep the distance the same: A will see B travel at 0.5c and cover the 10 ly to BC in 20 years. A will see C travel at 0.8c and cover the 10 ly from BC to AC in 12.5 years. B's proper time recorded between AB and BC will be 20 * sqrt(1-(0.5/c)^2) = 17.32 years C will then set it's time to 17.32 and cover the distance from BC to AC in an additional 12.5 * sqrt(1-(0.8/c)^2) = 7.5 years 17.32 + 7.5 = 24.82 24.82 < 32.5 Here, v<u but tau_C is still less than tau_A. One of us must have made a mistake somewhere, and as someone who wouldn't have understood any of the math I just typed if I'd read it this morning, I'd appreciate being corrected on any mistakes I make so I can avoid them going forward.

I believe this is true if the accelerating observer travels back at the same speed at which he leaves, but that doesn't seem like a requirement of the twin paradox.

AB is Clock B passing Clock A (or the observer leaving A), BC is Clock B passing Clock C (or the observer reaching the equivalent point in space and turning around), AC is Clock C passing Clock A (or the observer returning to A).

xyzt, would you say this statement is accurate: Adding the proper time measured by Clock B between events AB and BC to the proper time measured by Clock C between events BC and AC will yield a number that is equivalent to the proper time experienced by an observer that travels from event AB to event BC and then instantly accelerates to travel back to event AC?