Layman's Question about Special Relativity

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I'm just an undergraduate with a layman's interest in Physics. With regards to special relativity, I think I grasp the concept that the laws of physics are the same for all observes in uniform motion relative to one another. So if I am standing still and a spaceship zooms past me at 80% the speed of light, then I'm within my rights to say that relatively to me, the spaceship is moving at 80% the speed of light. However, there's nothing special about my state and the spaceship can say that it is standing still and that I am moving by at 80% the speed of light and that declaration is no less valid than my own.

Likewise, if I am driving down the highway at 50 mphs and a car in the opposite lane passes me and goes 60mph in the other direction, then by my measure, the other car is moving 110mph.

So what happens if I am in Spaceship A, moving along at 80% the speed of light and I meet Spaceship B that approaches and passes me in the opposite direction, also moving at 80% the speed of light. After Spaceship B passed me, if I were to measure his speed wouldn't it be 80% the speed of light + 80% the speed of light, thus exceeding the speed of light?
 

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  • #2
jbriggs444
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So what happens if I am in Spaceship A, moving along at 80% the speed of light and I meet Spaceship B that approaches and passes me in the opposite direction, also moving at 80% the speed of light. After Spaceship B passed me, if I were to measure his speed wouldn't it be 80% the speed of light + 80% the speed of light, thus exceeding the speed of light?
Velocity addition does not work that way under special relativity. You can Google "relativistic velocity addition" and see that velocities combine according to a rule of
$$v_{tot} = \frac{v_1 + v_2}{1+\frac{v_1v_2}{c^2}}$$
For automobiles going down the highway, the denominator is so close to one that it can be ignored. For spacecraft moving at significant fractions of the speed of light, it cannot be ignored.

One might wonder why velocities do not add directly. It seems blatantly against the rules of common sense that they would not. One way of addressing that concern is to note that it's like adding apples and oranges. You have ##v_1##, the velocity of some hypothetical space station relative to you. You have ##v_2##, the velocity of spacecraft B according to that space station. You consider adding these together directly. But they are velocities measured against different reference frames. They do not add directly. Instead, they "compose" or "boost". And for that, you need the formula above.

Edit: You can take your velocity of 0.8c and the other fellow's velocity of -0.8c and obtain a "closing velocity" of 1.6c -- the rate at which your mutual separation decreases as judged by a bystander on the space station. But that's not a relative velocity of any one thing in the rest frame of another.
 
  • #3
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No you have to use the relativistic velocity addition to figure it out.

You've also introduced a new observer one that sees both spaceship A and spaceship B moving at 80% the speed of light in opposite directions.

The relativistic velocity addition allows the spaceship observer to compute the speed of the other spaceship and it will be less than the speed of light.

https://en.wikipedia.org/wiki/Velocity-addition_formula

Here's a video that describes it:

 
  • #4
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No you have to use the relativistic velocity addition to figure it out.

You've also introduced a new observer one that sees both spaceship A and spaceship B moving at 80% the speed of light in opposite directions.

The relativistic velocity addition allows the spaceship observer to compute the speed of the other spaceship and it will be less than the speed of light.

https://en.wikipedia.org/wiki/Velocity-addition_formula

Here's a video that describes it:

Thank you for your response and I did check out the video. To be clear, I accept your conclusions and I do follow the formula provided, but I still don't understand WHY it is so. Or in response to the first jbriggs444, I understand that "it doesn't work that way", but I don't understand WHY it doesn't work that way. Also, I didn't mean to introduce a new observer. I only meant to consider the viewpoint of the person in Spaceship A, trying to measure the speed of Spaceship B without any consideration for a 3rd observer. I suspect the variable that I'm not considering has something to do with space or time dilation?

Let me try to understand with a similar example. Spaceship A is traveling at .9900c towards Spaceship B which is also traveling at .9900c towards Spaceship A. While Spaceship B is still many light years away I, sitting in spaceship A, decide to determine the speed of spaceship B. What result will I get? Then, light years later, the ships nearly hit head on, but miss and keep their constant speeds. Now Spaceship A is traveling at .9900c and Spaceship B is traveling at .9900c directly away from each other. Using the formula, I understand that, measuring from Spaceship A, Spaceship B is traveling at only .9999c. That's bizarre to me, but I understand that's the correct result.

Put another way, let's pretend that instead of a massive black hole, there is a planet in the exact middle of the milky way galaxy. Drawing a line through the poles of the planet, let's say Spaceship A moves at .9900c straight "north" or "up" while Spaceship B moves at .9900c straight "south" or "down" with each ship heading towards opposite edges of the galaxy. Just after launch inside Spaceship A, I look back at the planet. By my observation, I'm standing still and the planet is moving away at .9900c, right? (Likewise, an observer on the planet would say he was standing still and I was moving away at .9900c) So far so good?

Now, looking out my rear window, I decide to also measure the speed of SpaceShip B. According to the formula, SpaceShip B is moving at .9999c instead of my erroneous calculation of .99c+.99c. Okay.

Okay so here I am in spaceship A and a year goes by. By my calculation, I should find myself .9900 light years away from the planet. Likewise, by using the formula to calculate, Spaceship B should be .9999 light years away, just .0099 light years further than the planet. How can this be? During my "year" of travel inside Spaceship A, Spaceship B has been moving at .9900c in the opposite direction from the planet (relative to the planet). Yet, during my "year" of travel in the opposite direction from Spaceship B, I'm .99 light years away from the planet, while Spaceship B is only 0.0099 light years away from the planet in the opposite direction (or in other words spaceship B has only traveled 1% of the distance I have traveled in Spaceship A).

Is this because of time dilation? How much time has passed for Spaceship B during the year I spend recording time inside Spaceship A? Is it approximately 1/100th of a year?
 
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  • #5
jbriggs444
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I suspect the variable that I'm not considering has something to do with space or time dilation?
Time dilation, length contraction and relativity of simultaneity. The last is the one that most laymen miss.

You can't add a velocity from frame A to a velocity from frame B if your rulers aren't the same length, your clocks don't tick at the same rate and weren't synchronized the same way in the two frames.
 
  • #6
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I still don't understand WHY it is so.
The reason “why” for all relativistic effects is:

1) the principle of relativity
2) the invariance of c
 
  • #7
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Time dilation, length contraction and relativity of simultaneity. The last is the one that most laymen miss.
Would it be possible for you or someone to help me understand this in the context of my example scenario? In my last example, after the occupant in Spaceship A has experienced a year passing, how much time has gone by on the planet and how much time has gone by for the Spaceship B which is traveling .99c in the opposite direction from the planet?
 
  • #8
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The reason “why” for all relativistic effects is:

1) the principle of relativity
2) the invariance of c
Do you think that's helpful. Really?
 
  • #9
Grinkle
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@Peter Mole Post 6 is the answer. These two items are axioms from which the rest of the stuff is derived. Item 1 is intuitively appealing, and item 2 is experimentally verified. Niether 1 or 2 is, as far as I know, derived from more fundamental axioms.

I like the below thread on the twin paradox as good discussion for the question in your post 7.

https://www.physicsforums.com/threads/twin-paradox-and-the-body.940272/
 
  • #10
phinds
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Do you think that's helpful. Really?
Since it is the correct answer and ultimately explains everything in SR, I for one think it could not possibly BE more relevant. If you don't get that, don't blame Peter.
 
  • #11
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Since it is the correct answer and ultimately explains everything in SR, I for one think it could not possibly BE more relevant. If you don't get that, don't blame Peter.
Dude, I get it. No one here owes me anything. I'm just someone of average intelligence who didn't go much beyond high school algebra but who is also fascinated by relativity and quantum physics and so I'm trying to understand these fascinating concepts for no other reason than intellectual curiosity and I really do appreciate being able to engage with other people who are generous enough to share their understanding and share my wonder of it all.

That said, I'm not asking philosophical questions nor am I trying to challenge anyone's answer so much as I'm humbly trying to understand them. Jbriggs44 and others understood the context of what I was asking. Dale (not Peter, as I'm Peter) was, at best being just a little be snarky, and at worst, he was being condescending. His answer might as well have been "Because of Physics". And finally, I'm not trying to blame anyone for anything and I think it's strange that you inferred as much.
 
  • #12
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Dale (not Peter, as I'm Peter) was, at best being just a little be snarky, and at worst, he was being condescending
I was being neither snarky nor condescending. I was giving you a direct answer to the question you asked in the only way that science can answer such questions. I don’t know why you chose to interpret my motives so negatively. Of course, it is notoriously easy to misinterpret motives in this medium, so I suspect it was simply one such misunderstanding.

That said, a more complete explanation would have connected the postulates with the specific relativistic effect in question. But many people understand how to go from the postulates to the relativistic effect in question once it is pointed out to them where to start with any “why” question. So (being pressed for time) I chose to provide the simple answer immediately and a more complete answer later if you requested further details.
 
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  • #13
phinds
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... Dale (not Peter, as I'm Peter) was, at best being just a little be snarky, and at worst, he was being condescending.
I COMPLETELY disagree. He was giving the straightforward answer. Your reading attitude into it is your problem, not his.

EDIT: I see he beat me to it.
 
  • #14
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@Peter Mole

I like the below thread on the twin paradox as good discussion for the question in your post 7.

https://www.physicsforums.com/threads/twin-paradox-and-the-body.940272/
Thanks for the reply, but that thread doesn't seem to address my confusion. The OP in that discussion seems to struggling with the idea of how time can be moving normally for the traveler, before being corrected by Phinds later on down the thread. Likewise the accelerations aspects of that discussion seem to be more liken to general relativity where my questions are more solidly in the realm of SR.
 
  • #15
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Put another way, let's pretend that instead of a massive black hole, there is a planet in the exact middle of the milky way galaxy. Drawing a line through the poles of the planet, let's say Spaceship A moves at .9900c straight "north" or "up" while Spaceship B moves at .9900c straight "south" or "down" with each ship heading towards opposite edges of the galaxy. Just after launch inside Spaceship A, I look back at the planet. By my observation, I'm standing still and the planet is moving away at .9900c, right? (Likewise, an observer on the planet would say he was standing still and I was moving away at .9900c) So far so good?

Now, looking out my rear window, I decide to also measure the speed of SpaceShip B. According to the formula, SpaceShip B is moving at .9999c instead of my erroneous calculation of .99c+.99c. Okay.

Okay so here I am in spaceship A and a year goes by. By my calculation, I should find myself .9900 light years away from the planet. Likewise, by using the formula to calculate, Spaceship B should be .9999 light years away, just .0099 light years further than the planet. How can this be? During my "year" of travel inside Spaceship A, Spaceship B has been moving at .9900c in the opposite direction from the planet (relative to the planet). Yet, during my "year" of travel in the opposite direction from Spaceship B, I'm .99 light years away from the planet, while Spaceship B is only 0.0099 light years away from the planet in the opposite direction (or in other words spaceship B has only traveled 1% of the distance I have traveled in Spaceship A).

Is this because of time dilation? How much time has passed for Spaceship B during the year I spend recording time inside Spaceship A? Is it approximately 1/100th of a year?

Let's say mile-stones have been placed along the galaxy, one mile apart in the rest-frame of the galaxy. Let's say everything stays still in said galaxy, except for those two spaceships.

Ok, now pilots in both spaceships say: "The other pilot passes mile-stones at 1/70 of the rate that I pass mile-stones. And the mile-stones are 1/70 miles apart in this galaxy".

Time dilation factor and length contraction factor are both 1/70, when speed is 0.9999 c.


Oops, the mile-stones are 1/7 miles apart according to a pilot that is traveling at 0.99 c, relative to the mile-stones.
 
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  • #16
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I was being neither snarky nor condescending. I was giving you a direct answer to the question you asked in the only way that science can answer such questions.
At this point, as I mentioned, it seems to me you interpreted my question as philosophical or existential in nature. It wasn't, but I can see why such questions, particularly from bible thumpers and the like, might well grow tiresome and elicit a short dismissive reply. Likewise, I'm sure a lot of people simply come here perplexed by the ramifications of relativity and so they (mostly innocently) start wading into questions of why reality is the way it is instead of trying to understand how it works. It's a like the guy who wants to know WHY you can't know both the location and velocity of the electron at a specific instant in time. Again, these aren't the "why" questions I was asking, but I can see why you might have thought so and why a short dismissive answer might have seemed appropriate, but nevertheless, from my perspective, not very helpful.

I don’t know why you chose to interpret my motives so negatively. Of course, it is notoriously easy to misinterpret motives in this medium, so I suspect it was simply one such misunderstanding.
I really didn't interpret them so negatively. Perhaps I should have said "slightly glib" or "dismissive" on the one side of the scale instead of "snarky".

That said, a more complete explanation would have connected the postulates with the specific relativistic effect in question. But many people understand how to go from the postulates to the relativistic effect in question once it is pointed out to them where to start with any “why” question. So (being pressed for time) I chose to provide the simple answer immediately and a more complete answer if you requested further details.
Rather than point to mathematical theorems, I'm better off getting an understanding by dealing with concepts. So in my example where Spaceship A and Spaceship B are both approaching each other on a straight line each at .99c, what value for speed will the captain of Spaceship A calculate for Spaceship B? Thanks to jbriggs444 and jedishrfu, I have the formula for A & B moving away from each other, but it doesn't seem to work for A & B heading towards each other, although I'm certain I could be mistaken.

Beyond that I'll repeat my above question. Say observer A leaves earth at .99c going "up" and observer B stays on the planet, and observer C leaves earth at .99c doing "down" in the opposite direction of A. As I understand it, Observer B (on earth) can fairly state that both A and B are moving away from the earth at .99c. Likewise, Observer A can state that earth is moving away at .99c and Observer B can likewise state that earth is moving away at .99c. What I get hung up on is why, according to Observer A, observer C only appears to be moving away at .9999c (only .0099c greater). It has been explained to me this is because of time and space dilation which, at least in terms of time, I think I understand. Relative to Observer B (earth), time is moving slower for A and for C. However slower, I don't have the math to calculate, but it seems to me that the degree to which time is moving more slowly for A it would be slowing to the same degree for C, at least according to Observer B (earth). So if a year passes for A, then how much time is passing for C during the same "year" that passes for A? And for good measure, how much time would have passed for B (on earth), during A's year of time passing. (I ask so I can compare it to what has passed for C).

Is this all related to time/space dilation or is there something about the fact that A & C are moving in opposite directions that has a baring? I feel like there's more than just time/space dilation that I'm not accounting for.[/QUOTE]
 
  • #17
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I COMPLETELY disagree. He was giving the straightforward answer. Your reading attitude into it is your problem, not his.

EDIT: I see he beat me to it.
Good grief man, you didn't even get the posters name right so why are you looking to fight his battles? Go someplace else to look for an online fight. I'm not interested.
 
  • #18
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Let's say mile-stones have been placed along the galaxy, one mile apart in the rest-frame of the galaxy. Let's say everything stays still in said galaxy, except for those two spaceships.

Ok, now pilots in both spaceships say: "The other pilot passes mile-stones at 1/70 of the rate that I pass mile-stones. And the mile-stones are 1/70 miles apart in this galaxy".
Thanks for the reply. I'm a little confused if you are continuing with my example or starting fresh. So both pilots are traveling at .9999c? Relative to what? And they are traveling on such a path that at some point they will hit the first and then the second milestone? Does it matter if they are traveling one after the other along the path or if they are coming from opposite directions? I'm not trying to be pedantic, but I'm a little confused at the details or if the details I'm asking about are even important. Can you clarify a little more?

Time dilation factor and length contraction factor are both 1/70, when speed is 0.9999 c.


Oops, the mile-stones are 1/7 miles apart according to a pilot that is traveling at 0.99 c, relative to the mile-stones.
Thanks!
 
  • #19
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At this point, as I mentioned, it seems to me you interpreted my question as philosophical or existential in nature.
I assumed your question was scientific and so I answered it scientifically. My understanding from your stated background is that you have no formal training in relativity, but a curiosity. You understand what the effect is, but not why it occurs.

The scientific reason why is because of the two postulates. If you can accept those two statements as true, then all the other relativistic effects logically follow. As a student, when you get stuck in relativity, it is nice to have a firm footing that you can always go back to and proceed from. Those postulates are that firm footing, the answer to all “why” questions in relativity.

Perhaps I should have said "slightly glib" or "dismissive"
It wasn’t dismissive. You asked a question which had a direct and brief answer, which I provided. It was brief, but with 5 kids and a homestead sometimes brief is all I can do. I also prefer brief answers whenever possible as there is less room for confusion.

Rather than point to mathematical theorems, I'm better off getting an understanding by dealing with concepts.
Well, concepts and mathematical theorems are pretty close in relativity. The problem is that this is a realm of the universe where instinct and intuition are fairly useless. Our caveman ancestors did not need to factor in relativity when throwing spears at a wolly mammoth, so our brains did not evolve over millennia to instinctively understand these concepts. So instead we rely on math to guide us and then on experiments to check our math.

In a conceptual way, from the two postulates you can obtain the Lorentz transforms, and then from the Lorentz transforms you can obtain the relativistic velocity addition.

It has been explained to me this is because of time and space dilation which, at least in terms of time, I think I understand.
Well, it is because of the Lorentz transform, which includes time dilation, length contraction, and also the relativity of simultaneity. The latter effect is the one that most frequently trips new students. It is also the effect which is most difficult to grasp. This is what you are missing here.

I feel like there's more than just time/space dilation that I'm not accounting for.
Yes, the relativity of simultaneity.

It is easier to use the Lorentz transform than it is to piecemeal use length contraction, time dilation, and relativity of simultaneity. Are you familiar with the Lorentz transform?
 
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  • #20
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Beyond that I'll repeat my above question. Say observer A leaves earth at .99c going "up" and observer B stays on the planet, and observer C leaves earth at .99c doing "down" in the opposite direction of A. As I understand it, Observer B (on earth) can fairly state that both A and B are moving away from the earth at .99c. Likewise, Observer A can state that earth is moving away at .99c and Observer B can likewise state that earth is moving away at .99c. What I get hung up on is why, according to Observer A, observer C only appears to be moving away at .9999c (only .0099c greater).
But this is precisely why @Dale's explanation was correct in the first place: it would violate the second postulate if observers A or C could measure the other moving at or faster than c. The invariance of c makes it a universal speed limit—no matter how hard you try to "catch up" to a light wave, its speed remains the same. So even before doing the math to figure out the details, you know that A and C must measure each other moving slower than c.

Say that light is emitted in the "down" direction from "above" observer A. Observer A says it's moving at c, but A also knows that if he switched to C's reference frame the light would still be moving "down" at c, even though C is moving very fast in the same direction as the light from A's perspective. If switching to C's reference frame leaves the light's velocity unchanged, then C must be moving slower than the light in A's frame, and indeed in all frames.
 
  • #21
Grinkle
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that thread doesn't seem to address my confusion
Fair enough. I am not sure if you are confused about how to do a specific calculation or about something more fundamental. So acknowledging that I am not confident my further comments are helpful for you, I offer them anyway for fwiw.

It helped me a lot to take the two axioms separately. The principle of relativity leads to two consistent models of the universe - Galilean invariance and Lorentz invariance.

Edit - hit save too soon.

If you can follow the arguments that either of these systems (and only these systems) yield a universe in which the principle of relativity holds, then it is a small jump to accepting the experimental evidence that our Universe is in fact Lorentz invariant. It needn't be, for the principle of relativity alone to be true, but it is.

Then the rest of the details in the problem you are asking about are math - not really intuitive (at least not at all for me) until one has deliberately trained the intuition by working out the math on many such problems.
 
  • #22
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I assumed your question was scientific and so I answered it scientifically. My understanding from your stated background is that you have no formal training in relativity, but a curiosity.
yep

You understand what the effect is, but not why it occurs.
Right, I was trying to understand if the effect was a matter of time/space dilation or if it somehow had to do with the direction or distance in space itself or by some other misunderstanding I had about how relativity worked. I'm hoping to better understand through examples that show how much time has passed for the observers inside the senarios I have asked about.

The scientific reason why is because of the two postulates. If you can accept those two statements as true, then all the other relativistic effects logically follow. As a student, when you get stuck in relativity, it is nice to have a firm footing that you can always go back to and proceed from. Those postulates are that firm footing, the answer to all “why” questions in relativity.

It wasn’t dismissive. You asked a question which had a direct and brief answer, which I provided. It was brief, but with 5 kids and a homestead sometimes brief is all I can do. I also prefer brief answers whenever possible as there is less room for confusion.
Nonsense. Scroll back up an look at my unedited OP. Flawed as my understanding may be, my question clearly demonstrated that I was familiar with the theory of relativity. In fact, I explicitly mentioned it an even alluded to the definition. Furthermore, the nature of my question clearly shows that I was aware there was a problem with going faster than the speed of light. Clearly, if I wasn't aware of the theory of relativity and the constant speed of light, I wouldn't have even been able to form the question.

Yet, when I asked "why" at one point, you decided the correct response was to point me to the theory I was already using, breaking it up into two postulates that my question demonstrated I was already aware of. You might as well have answered, "Because of Physics". Perhaps your household duties and children meant you didn't have the time to read my OP.

Well, concepts and mathematical theorems are pretty close in relativity. The problem is that this is a realm of the universe where instinct and intuition are fairly useless. Our caveman ancestors did not need to factor in relativity when throwing spears at a wolly mammoth, so our brains did not evolve over millennia to instinctively understand these concepts. So instead we rely on math to guide us and then on experiments to check our math.
Yep, I'm still aware of the concept of relativity and the fact that it doesn't follow instinct and intuition, as my OP clearly demonstrates. Also, having a layman's understanding of relativity (as my OP clearly shows), of course I'm aware that mathematics plays a crucial role, although I don't claim to have a good grasp of the math which is why I'm trying to understand it conceptually.

In a conceptual way, from the two postulates you can obtain the Lorentz transforms, and then from the Lorentz transforms you can obtain the relativistic velocity addition.

Well, it is because of the Lorentz transform, which includes time dilation, length contraction, and also the relativity of simultaneity. The latter effect is the one that most frequently trips new students. It is also the effect which is most difficult to grasp. This is what you are missing here.

Yes, the relativity of simultaneity.

It is easier to use the Lorentz transform than it is to piecemeal use length contraction, time dilation, and relativity of simultaneity. Are you familiar with the Lorentz transform?
Okay thanks. Relativistic velocity addition was already mentioned to me as was relativity of simultaneity. I'll add Lorentz transform to my list of things to research. I'm not smart enough absorb these ideas without investing a lot of time. Maybe I can utilize these theorms to answer the specific questions I've asked here in the thread which would go a long way to helping me understand what's actually happening in terms of time/space dilation, which is all I'm really interested in anyway.
 
  • #23
ZapperZ
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So, this is the reason why I tried to highlight MinutePhysics' current "project" in "teaching" SR.

https://www.physicsforums.com/threads/minutephysics-special-relativity-series.938686/

Many of the issues you brought up in your original post can be found in at least a couple of videos listed in that thread. Even Fermilab's Don Lincoln is doing his own series on SR, and his videos are also included in the thread.

Maybe you might want to start with those. And as to a direct answer to your original post, the answer is no, you can't add velocities that way. The way we commonly add velocities is called Galilean transformation. It turns out, as I've pointed out in the Insight article, this is simply an approximation that is valid for v<<c. The more GENERAL description, as you've been told, is the Lorentz transformation. If you had done a search on "relativistic velocity addition", you would have found several websites illustrating to you how this is done.

So before moving on further in the "why's", which seems to be what you are asking in your subsequent posts, can we first get THIS ONE clarified to your understanding?

Zz.
 
  • #24
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Nonsense. Scroll back up an look at my unedited OP. Flawed as my understanding may be, my question clearly demonstrated that I was familiar with the theory of relativity. In fact, I explicitly mentioned it an even alluded to the definition. Furthermore, the nature of my question clearly shows that I was aware there was a problem with going faster than the speed of light. Clearly, if I wasn't aware of the theory of relativity and the constant speed of light, I wouldn't have even been able to form the question.

Yet, when I asked "why" at one point, you decided the correct response was to point me to the theory I was already using, breaking it up into two postulates that my question demonstrated I was already aware of. You might as well have answered, "Because of Physics". Perhaps your household duties and children meant you didn't have the time to read my OP.
#DaleDidNothingWrong
 
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Say that light is emitted in the "down" direction from "above" observer A. Observer A says it's moving at c, but A also knows that if he switched to C's reference frame the light would still be moving "down" at c, even though C is moving very fast in the same direction as the light from A's perspective. If switching to C's reference frame leaves the light's velocity unchanged, then C must be moving slower than the light in A's frame, and indeed in all frames.
Yep, I understand that the speed of light is constant and that it cannot be exceeded.

Let me just ask the same question again and see if someone can help give me mathematically answers or point me to mathematical equations to get my own answers, although I'm not well versed in much beyond high school algebra.

Three positions. A, B, C, all starting at the same point in space. Next, by B's observation, A instantly begins to move west at .9900c. Also by B's observation, C instantly begins to move east at .9900c.

By A's observation, A is moving at .9900c relative to B. By C's observation, C is moving at .9900c relative to B.

EDIT: (thanks jbriggs444) I meant to say By A's observation, B is moving away at .9900c relative to A. Also, by C's observation, B is moving away at .9900c relative to C.

Using the relativistic velocity addition provided above, I know that by A's observation, C is moving at .9999c, and that by C's observation, A is moving at .9999c.

Now, staying with A's frame of reference, a year passes. Relative to B, A has been moving at .9900c for a year. Although I don't know how to do the calculation, I understand that much less time has passed from B's perspective during the time that a year has passed for A. My question is, how much time has passed for C, and how far away is C from A at the moment that a year has passed for A's inside A's frame of reference. If the math is too involved to give me specific answers of time and distance, just knowing approximate answers might be enough to bridge my understanding of what's going on.
 
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