Time Dilation & Length Contraction, Further Thoughts

[ghwellsjr]

It was confirmed in post 22 that everything can consider itself at rest in its own frame and all frames are equivalent.
Light moves at a constant speed so why can it not be considered at rest in its own frame. The absurd results are presumably that time dilates to infinity and length contracts to 0, this seems no more absurd than a singularity, either within a black hole or at the BB.

Look at the spacetime diagrams in post #12 in which the Earth is at rest. It is shown as a vertical blue line. The light signals that are sent from the Earth to the muon are shown as yellow lines going up and to the right at a 45 degree angle.

Now look at the spacetime diagrams in post #13 in which the muon is at rest. It is shown as a thick vertical black line. The light signals that are sent from the muon to the Earth are shown as thin black lines going up and to the left at a 45 degree angle.

Any object that is at rest in an IRF will be shown as a vertical line. All light signals will be shown as lines at a 45 degree angle. How do you propose to show light at rest in its own frame in which it must be drawn as a vertical line at a 45 degree angle?

The equations just show that time and space are linked, as one expands the other contract and just that the limits seem to be 0 and infinity. I can see no real problem, velocity is just time and distance, if one changes then the other has to in order to maintain the constant speed of light, obviously the limits cannot go to 0 or infinity therefore nothing can be absolutely at rest so as we have no way of knowing the true actual speed of anything in the universe we just have to assume the observer is at rest in his own frame and go from there. We assume ourselves to be at rest within our own frame but we are moving at considerable speed through space, there can be no actual or true rest frames on frames at rest relative to others i.e a car (A) moving at 60mph is not at rest but compared to a car (B) moving at 120 mph its occupants have a relative speed of 60mph if they consider themselves at rest ( having no means of knowing whether or not they are moving, they also have a relative speed of 60 compared to a car ( C )that is not moving.
The occupants of car C would get a Time dil. and length con of x for car A and y for car B, the occupants of car A would get x for both car B and C this gives different results for the same cars moving at constant speed relative to each other, car A must get incorrect results because they assume they are at rest when they are not.

As Doc Al pointed out, you have to use the correct forumula to add (or subtract) speeds. I have already shown you spacetime diagrams in which the Earth is at rest and in which the muon is at rest. Now I want to show you another spacetime diagram for an IRF in which both the Earth and the muon are traveling at the same speed. This would be kind of like the rest frame for car (A) except that there is nothing at rest in this IRF and instead of traveling away from each other as the cars are, the Earth and muon are traveling toward each other.

The common speed of the Earth and the muon in this IRF is not just one-half of their relative speed of 0.999c. Instead it is 0.95625c. You can plug this into both inputs of the Velocity Addition formula (see wikipedia) and verify that they add to 0.999c. The value of gamma at 0.95625c is 3.41821. Here is the diagram:

Note first off that both the Earth and the muon are time dilated by the same amount. You can see that 2 microseconds for the muon, from event C to event E is the same coordinate interval as 2 microseconds for the earth, say from its Proper Time of 84 to 86.

Note secondly that the distance between the Earth and the muon at the time of the muon's creation in this IRF is about 13 light-microseconds as depicted by the distance from event F to event C. Since both the Earth and the muon are traveling at 0.95625c, they each cover half that distance and meet at event E. It takes them each 6.83642 microseconds of coordinate time to travel that coordinate distance but since they are both time dilated by the same amount, it is only 2 microseconds of their own Proper Time.

Also note that the light signals that each one sends and receives is exactly the same in this IRF as it is in the two previous IRF's. That's an important lesson to learn. There is no difference in terms of what observers measure or see when described by different IRF's that are transformed from one to the other. You can still count the same number if yellow signals received by the muon during its trip in this IRF as are in the previous two IRF's. The muon sends out two black signals to Earth which receives them about half way between its Proper Time of 89 and 90 microseconds. None of these features could possibly work with your idea that light could have it own rest frame.

One more thing: you said that the equations show that time and space are linked, as one expands the other contracts but if you look at the three sets of diagrams, you will see that as we transform to different IRF's the coordinates for events tend to all exand or contract together. I think you need to study these diagrams and realize that they are all simply derived from the coordinates of events in one IRF transformed into different coordinates for other IRF's.

 

[Adrian07]

I think I know where we have our lines crossed.
I have been trying to reconcile the following
! A rest frame of some object is a reference frame in which the object's velocity is zero.

2 One of the key axioms of special relativity is that light moves at c in all reference frames. The rest frame of a photon would require the photon to be at rest (velocity=0) and moving at c (velocity=299792458 m/s). That of course is contradictory. In other words, the concept doesn't make sense.

3 Everything can consider itself at rest in its own frame and all frames are equivalent.

4 The speed of light is constant.

5 Time dilation Length contraction.

1 makes little sense as nothing has 0 velocity, we only have 0 velocity relative to our immediate surroundings we just set our velocity to 0 in relativity and do so for all IRFs whatever their actual velocity
I think you are using classical interpretation of relativistic numbers. I can see no problem with a photon being at rest and moving at c, this is precisely what 5 tells us is happening. It is at rest in its own frame (relativity) and moving at c relative to us (classical). Speed is in time and distance units, to make it easy we will use 1, Second, time unit and 1 distance unit, we split this into meters, this gives the speed of light as 1/1. This is relative to us and assumes the units are constant and unchanging.
But 5 tells us that these units are subject to change and how they change, so the speed of light is relative to the actual speed of the frame from which it is measured, so for a time dilation factor of 2 we get 2/.5 in that frame but still remaining as 1/1 relative to us, this seems to say that with increasing speed the relative speed of light decreases by Time dil over length cont. but will appear to stay constant because it will always be 1 time unit/ 1 distance unit in each frame of reference.
In 2 above you have moved our units of time and length into an IRF moving at c without applying relative time dil and length cont which would show the photon al rest in its own frame.

 

[Dale]

I can see no problem with a photon being at rest and moving at c

That is a fairly silly comment. If v = c and v = 0 then 299792458=0 which is clearly a false statement. If you can see no problem with that then a brief review of the meaning of = would be appropriate.

 

[ghwellsjr]

I think I know where we have our lines crossed.
I have been trying to reconcile the following
! A rest frame of some object is a reference frame in which the object's velocity is zero.

2 One of the key axioms of special relativity is that light moves at c in all reference frames. The rest frame of a photon would require the photon to be at rest (velocity=0) and moving at c (velocity=299792458 m/s). That of course is contradictory. In other words, the concept doesn't make sense.

3 Everything can consider itself at rest in its own frame and all frames are equivalent.

4 The speed of light is constant.

5 Time dilation Length contraction.

1 makes little sense as nothing has 0 velocity, we only have 0 velocity relative to our immediate surroundings we just set our velocity to 0 in relativity and do so for all IRFs whatever their actual velocity

OK, we'll make a new rule, just for you. We'll call it Adrian's special Special Relativity. We will disallow any IRF in which anything is traveling at 0 velocity. Note that I just made you a diagram for an IRF in which nothing is at rest, not the earth, not the muon and not the light. In fact, I will make you a new diagram for an IRF in which the Earth is traveling at its "actual velocity". Just tell me what that velocity is, anything between 0.001c and 0.998c, and I'll do it.

I think you are using classical interpretation of relativistic numbers. I can see no problem with a photon being at rest and moving at c, this is precisely what 5 tells us is happening. It is at rest in its own frame (relativity) and moving at c relative to us (classical).

According to your new rule that "nothing has 0 velocity", we disallow any IRF in which light is at rest, so this is a meaningless issue.

Speed is in time and distance units, to make it easy we will use 1, Second, time unit and 1 distance unit, we split this into meters, this gives the speed of light as 1/1. This is relative to us and assumes the units are constant and unchanging.
But 5 tells us that these units are subject to change and how they change, so the speed of light is relative to the actual speed of the frame from which it is measured, so for a time dilation factor of 2 we get 2/.5 in that frame but still remaining as 1/1 relative to us, this seems to say that with increasing speed the relative speed of light decreases by Time dil over length cont. but will appear to stay constant because it will always be 1 time unit/ 1 distance unit in each frame of reference.

2/.5 = 1/1?? Remember, we cannot measure the speed at which light propagates. Einstein assigns its speed to be c in any IRF. All we can measure is the round-trip speed of light and it won't matter what IRF we use to illustrate that.

In 2 above you have moved our units of time and length into an IRF moving at c without applying relative time dil and length cont which would show the photon al rest in its own frame.

I don't recall moving our units of time and length into an IRF moving at c. Can you show me the post where I did that? And you have already declared that "nothing has 0 velocity" so why do you continue to talk about a photon at rest in its own frame?

 

[Adrian07]

Sorry about the controversy and I think I may need to tread carefully as I don't want to get into trouble with the administrators. I also wonder if thinking about quantum theory may also help.

Regarding the photon at rest. The speed of light is constant it is the units used to measure it that change, you are moving meters relative to us into a frame moving at c.
I have a clock and a measuring stick 300,000,000 mts long, light will always travel the length of my stick in one tick of my clock. As I move faster the stick contracts and the clock ticks slower but light will always travel one length in one tick, you can obviously see where this ends up. However I think that as length contracts to 0 quantum theory must come into play especially the uncertainty principle in order to remove singularities and infinities. I think I am right in saying the uncertainty principle forbids anything from being totally at rest so the photon can never have 0 velocity so perhaps the uncertainty in its speed scaled up using relativity = what we know as the speed of light.

Dont know if this is relevant but the muon being a quantum particle must obey quantum rules so by saying where it is and how fast it is moving in the spacetime diagrames are you not breaking the uncertainty principle by saying exactly where it is and how fast its moving.

The fact is that light moves in all reference frames and everything can consider itself at rest in its own frame are contardictory.

I was out walking thinking about what we have been saying and how I look at the world. It occurred to me that if I had a meterstick and held it at arms length a tree in the distance would seem the same size as my stick, as I walk towards the tree it appears to get bigger relative to my stick, yet to someone standing next to the tree with a measuring stick my stick would appear to be getting bigger, of course anything I was moving away from would have the opposite effect. Is this not a simple example of what we have been talking about?
I am not trying to be controversial or say that accepted theories are wrong I am just saying what they are telling me.

 

[Dale]

Regarding the photon at rest. The speed of light is constant it is the units used to measure it that change, you are moving meters relative to us into a frame moving at c.

Length contraction and time dilation are irrelevant to the logical contradiction of a rest frame for light.

In order for a frame to qualify as the rest frame of light you need to have v=0 and v=c as measured in that single frame where v is the speed of a single pulse of light. There is no length contraction nor time dilation within a single frame, so the clocks and rods used to measure v=0 and v=c are the same.

It is a clear logical self-contradiction. The concept of a rest frame for light is nonsense, regardless of how appealing you found it while walking around the other day.

 

[ghwellsjr]

Sorry about the controversy and I think I may need to tread carefully as I don't want to get into trouble with the administrators. I also wonder if thinking about quantum theory may also help.

No, it won't help. Are you interested in learning Special Relativity? That's what this forum is for. You're not going to get into trouble with the administrators if you show evidence that you are focused on trying to learn about SR instead of ignoring every response and launching into yet another diversion. This is exactly what happened in your last thread that got locked.

Regarding the photon at rest. The speed of light is constant it is the units used to measure it that change, you are moving meters relative to us into a frame moving at c.
I have a clock and a measuring stick 300,000,000 mts long, light will always travel the length of my stick in one tick of my clock. As I move faster the stick contracts and the clock ticks slower but light will always travel one length in one tick, you can obviously see where this ends up. However I think that as length contracts to 0 quantum theory must come into play especially the uncertainty principle in order to remove singularities and infinities. I think I am right in saying the uncertainty principle forbids anything from being totally at rest so the photon can never have 0 velocity so perhaps the uncertainty in its speed scaled up using relativity = what we know as the speed of light.

So now you have another reason to believe that a photon cannot be at rest.

Dont know if this is relevant but the muon being a quantum particle must obey quantum rules so by saying where it is and how fast it is moving in the spacetime diagrames are you not breaking the uncertainty principle by saying exactly where it is and how fast its moving.

So what if I am? It is immaterial to learning SR. The point is that I can say, "If the muon comes into existence two light-usecs above the surface of the Earth in the Earth's IRF and travels straight down at 0.999c," then I can legitimately draw a spacetime diagram of that activity and I can legitimately use the Lorentz Transformation process to draw another spacetime diagram for the muon's IRF. That's what SR is all about and what you need to focus on if you want to avoid the wrath of the administrators.

And please don't ignore my previous question to you: what is the "actual velocity" of the earth? I need to know so that I can draw a spacetime diagram for an IRF that you will accept where nothing is at rest.

The fact is that light moves in all reference frames and everything can consider itself at rest in its own frame are contardictory.

Only if you insist on including photons [at rest] in that statement--so don't do that.

I was out walking thinking about what we have been saying and how I look at the world. It occurred to me that if I had a meterstick and held it at arms length a tree in the distance would seem the same size as my stick, as I walk towards the tree it appears to get bigger relative to my stick, yet to someone standing next to the tree with a measuring stick my stick would appear to be getting bigger, of course anything I was moving away from would have the opposite effect. Is this not a simple example of what we have been talking about?

It is a simple example of part of what we're talking about but not all.

I am not trying to be controversial or say that accepted theories are wrong I am just saying what they are telling me.

In this thread, you have focused on Doc Al's agreement with you in post #22 that "everything can consider itself at rest in its own frame" and you insist on including a photon as one of those things and yet you ignore Doc Al's earlier statement in post #2 of your other thread that light always travels at c. It doesn't appear to me that you are not trying to be controversial. If that were the case, it should have taken only one response for you to drop this notion that a photon can be at rest in its own frame.

So now the question is: are you going to keep on being controversial by insisting that a frame can travel at c so that a photon can be at rest in its own frame or are you going to drop that notion and learn Special Relativity? If you choose the former, you have only yourself to blame for the consequences that are obvious to you.

 

[Adrian07]

I have tried re-reading the thread to work out where my confusion lays. I think it is where you seem to be jumping from one frame of reference to another i.e. from the mnons to Earth's and back again so there is no control reference to compare with.
Hopefully you can see where my line of thought is leading me and explain simply where I am going wrong.
I will try another scenario using my clock and measuring stick as above.
We are both standing in the same frame which we use as a control and use the distance between the Earth and the sun as a reference for distance.
I will zoom off towards a hydrogen atom until my measuring stick, using relative length contraction, tells me there is 93 million miles between the proton and the electrons orbit, that is the same number of my sticks between the proton and electron as yours between the Earth and the sun. My clock will tell me that it takes 8 minutes for light to travel between the proton and electron, the same as yours between the sun and earth, you, in the control frame, will measure the same but as taking a fraction of a second to move a fraction of a meter thus the speed of light remains the same but our measurement of time and distance change. As we go smaller and smaller relatively the time and distance units become to small to measure and so appear to come to rest relative to the control frame, so the term at rest must also be treated relatively and is dependant on what can be measured thus in a photons frame it is at rest relative to the control frame because the time and distance units are to small to measure.
Please explain as simply as possible where I have misinterpreted the transforms. If the speed of light remains at 300,000,000mts per second in all frames as per the control frame my time dil. and length con. measuring instruments will give a completely different number.

 

[PeterDonis]

I will zoom off towards a hydrogen atom until my measuring stick, using relative length contraction, tells me there is 93 million miles between the proton and the electrons orbit, that is the same number of my sticks between the proton and electron as yours between the Earth and the sun.

This will never happen. The number of your measuring sticks between the proton and electron will get *smaller*, not larger, as you move faster and faster relative to the hydrogen atom. Your sticks don't shrink in your frame as you move; the atom does.

 

[PeterDonis]

I have tried re-reading the thread to work out where my confusion lays.

I think your confusion is that you keep trying to analyze things using only length contraction, or only time dilation. You can't do that. Length contraction and time dilation are not fundamental; they are emergent effects. To correctly analyze scenarios in relativity, it's best to start from the fundamentals. The best way I know to do that is spacetime diagrams, and ghwellsjr has already given you several good ones in this thread. Spacetime diagrams like the ones he has given you focus attention on events and their geometric relationships; once you have those correct, it's easy to derive answers to whatever questions you might have about length contraction and time dilation. But length contraction and time dilation are only part of the picture; spacetime diagrams give you the whole picture, and only the whole picture will give you the proper understanding of what's going on.

 

[Dale]

in a photons frame it is at rest relative to the control frame because the time and distance units are to small to measure.

There is no such thing as a photons frame for the reasons already explained over and over. It is a logical self-contradiction.

Please explain as simply as possible where I have misinterpreted the transforms.

c≠0
I don't think it can be explained more simply than that.

 

[ghwellsjr]

I have tried re-reading the thread to work out where my confusion lays. I think it is where you seem to be jumping from one frame of reference to another i.e. from the mnons to Earth's and back again so there is no control reference to compare with.

Maybe if you would answer my question ( this is the third time I've asked), you would accept an IRF as a control reference: What is the "actual velocity" of the earth?

Hopefully you can see where my line of thought is leading me and explain simply where I am going wrong.
I will try another scenario using my clock and measuring stick as above.
We are both standing in the same frame which we use as a control and use the distance between the Earth and the sun as a reference for distance.
I will zoom off...

There, I think you have identified your problem. You're always zooming off. Just answer my question above and I will make you an IRF that you can accept as a legitimate control reference. Otherwise, you can zoom off all you want but I'm staying behind.

 

[Adrian07]

The spacetime diagrams are fine its the explanation of them that is confusing and you are assuming that my level of knowledge and terminology are the same as yours.
As I am sure you know it is impossible for us to work out the actual velocity of anything, best guess for the Earth is between 600 and 650 kms/sec but that is relative to the great attractor and that is probably moving in an unknown direction at an unknown speed.
Are you trying to tell me, post 39, that length contraction means the thing I am moving toward is getting smaller? I will have to remember that as I move towards something the impression that its getting bigger is an optical illusion and things are actually getting smaller.
Getting back to the issue nobody has explained where my scenario is wrong. I have dilated time and contracted space and kept the speed of light constant in all frames, you tell me its wrong but not why.
Perhaps someone can define simply the term rest frame it seems to mean a frame that is not accelerating but is none the less not strictly speaking at rest, so the observer is at rest relative to himself, so how does this work though if the observer is a photon?
The reasons given so far are that it is a logical self contradiction, I can see no more contradiction than considering myself at rest knowing full well that without anything else, the Earth's rotation shows I am not, so standing on the equator I am at rest and moving at 1000mph is just as much a logical contradiction.
I realize you are probably getting frustrated with me but I am still unable to reconcile the comments on this thread with what the transforms seem to tell me. Not to mention the use of relativity in things like the gps system.
c≠0 please explain what this means and how it relates to my senario.
Please I need clear concise simple explanations, the spacetime diagrams are lines on a graph the explanations to me are neither clear nor concise.

 

[Dale]

Perhaps someone can define simply the term rest frame

The "rest frame of X" is defined as a reference frame where the speed of X is 0.

c≠0 please explain what this means and how it relates to my senario.

The speed of a light pulse is c in all reference frames. c≠0. Therefore there is no frame where the speed of a light pulse is 0. Therefore there is no rest frame of a light pulse.

I don't think it can possibly be more clear or concise than that.

 

[laurub]

allthough this thread did teach me a few new things, I can't seem to shake the suspicion that Adrian07 is a troll.
I'm sorry if I'm wrong, but there's just no way to be certain.

 

[ghwellsjr]

The spacetime diagrams are fine its the explanation of them that is confusing and you are assuming that my level of knowledge and terminology are the same as yours.
As I am sure you know it is impossible for us to work out the actual velocity of anything, best guess for the Earth is between 600 and 650 kms/sec but that is relative to the great attractor and that is probably moving in an unknown direction at an unknown speed.
...
Please I need clear concise simple explanations, the spacetime diagrams are lines on a graph the explanations to me are neither clear nor concise.

Let's see if we can bring up your level of knowledge and terminology. Since you don't have any problem with the spacetime diagrams, we'll work on the explanation.

First off, DaleSpam just gave you a definition of what a rest frame is. So even if the "actual velocity" of the Earth is known to be non-zero, we can still draw a diagram in which its speed is zero because we assume that the frame is moving right along with the Earth at whatever speed it is traveling at.

I have redrawn my previous spacetime diagram for the rest frame of the Earth from post #12 down to the bare minimum. Please explain what you think these lines and dots on the graph mean. Look back at post #12 if you want some hints. Tell me everything you know about this diagram as if you were explaining it to someone who doesn't understand anything about it at all:

 

[Adrian07]

The graph appears to show something starting at a point in time, not sure about a usec, moves backwards in time 45 usecs and forward through space 45 light secs.

One last question about rest frames, squrt 1 - v2/c2 tells us that at c T becomes infinite and length 0. please explain how something taking infinite time to move 0 distance is not at rest.

 

[Adrian07]

Laurub suggest if you have nothing sensible to add to the conversation you keep your comments to yourself.

 

[laurub]

The graph appears to show something starting at a point in time, not sure about a usec, moves backwards in time 45 usecs and forward through space 45 light secs.

a graph isn't allways read left to right, but in this case, right to left, so, something starts at 45 light-μseconds (light-microseconds), then, it moves in negative direction along the x-axes over a period of 0,000045 sec (or 45 μsec).

One last question about rest frames, squrt 1 - v2/c2 tells us that at c T becomes infinite and length 0. please explain how something taking infinite time to move 0 distance is not at rest.

in my understanding, v can never be c, but only approach c in a limit (please correct me if I'm wrong), so your question becomes irrelevant.

 

[ghwellsjr]

The graph appears to show something starting at a point in time, not sure about a usec, moves backwards in time 45 usecs and forward through space 45 light secs.

The unit "usecs" is just a common way of stating "μsecs" (for when you don't have Greek letters available) which is short for microseconds or millionths of a second. Does that help?

When drawing diagrams in Special Relativity, it is common to put time on the vertical axis rather than the more common horizontal axis. The labels on the graph show that the horizontal axis is for light-microseconds which is a unit of distance, not time. The blue line is showing the position of the Earth as a function of time. Since this IRF is for the rest frame of the earth, I show it at a position of 0 starting at a time of 0 in the bottom left corner of graph.

As time goes by, going up the graph, the Earth stays put but at about 45 microseconds from the start of the scenario, a muon is created high in the atmosphere, at an altitude of about 45 light-microseconds. A light-microsecond is the distance light travels in a microsecond which is about a thousand feet. So 45 light-microseconds is 45 thousand feet. since there are about five thousand feet in a mile, this is about 9 miles high in the atmosphere.

The muon travels straight down at 99.9% of the speed of light (shown in as a thick black line) and so it covers the distance of 45 light-microseconds in slightly over 45 microseconds and meets the surface of the Earth at about 89.5 microseconds into the scenario. These times and distance are only approximate so don't worry about the slight inconsistencies. Also don't worry at this point about the Proper Time indicated by the dots on the path of the muon.

So do you see that the muon is moving down and going forward in time?

One last question about rest frames, squrt 1 - v2/c2 tells us that at c T becomes infinite and length 0. please explain how something taking infinite time to move 0 distance is not at rest.

When we show the trace of a photon on a diagram, it will progress at exactly c so what ever distance it covers will be the same value for the time it takes. So look back at the first diagram in post #12. You see a yellow line starting at the coordinates of t=0 and x=0 and going up and to the right at a 45 degree angle. So at t=45 usecs, x=45 light-usecs. Now we use the Lorentz Transformation process to see what the coordinates of those two events are in another IRF moving at some speed with respect to the first one.

The speed, β, we are going to use is -0.999c. Gamma, γ, at this speed is 22.366272. The formulas for the new coordinates are:

t' = γ(t-βx)
x' = γ(x-βt)

For the first event at t=0, x=0, this comes out the same, t'=0, x'=0.
For the second event at t=45, x=45, it comes out to t'=2011.958, x'=2011.958.
You can see this on the diagrams in post #13.

I think you can see from the two formulas that if t=x, then t'=x' for any value of β no matter what the value is or how close to one it gets. And we can observe that the closer β gets to one, the larger t' and x' will get. So even if we want to see what happens as β gets "infinitely" close to one which is what happens when v gets infinitely close to c, the two values of t' and x' will still be equal to each other and the event they define is simply a very long way off on a 45 degree angle on the diagram. So the path of the photon is always a 45 degree angle on any IRF drawn with the t and x axes to the same scale.

Does this make sense to you?

 

[Nugatory]

One last question about rest frames, squrt 1 - v2/c2 tells us that at c T becomes infinite and length 0. please explain how something taking infinite time to move 0 distance is not at rest.

The time dilation and length contraction equations actually tell us no such thing, because they do not apply when v=c. They are derived using a sequence of mathematical steps that starts with the (usually implicit) statement "If v<c, then we can show that..." and proceeds from there.

Consider the formula for the area of a circle: A=\pi{r}^2. It's pretty clear that if I shrink the radius towards zero, the area of the circle shrinks to zero as well. But suppose I were to keep shrinking the radius to even smaller (negative) numbers... I could say that the formula "tells us" that the area of the circle starts to increase as the radius shrinks beyond a certain point. But just about everyone reading this (including you, I hope) would rightly tell me that the radius of a circle is always non-negative so I just plain can't attach any meaning to the area of a circle of negative radius.

 

[Dale]

One last question about rest frames, squrt 1 - v2/c2 tells us that at c T becomes infinite and length 0. please explain how something taking infinite time to move 0 distance is not at rest.

Again, there is no reference frame moving at c. See post 36.

You keep trying to start your reasoning from a false premise, and then seem surprised that you wind up with problems.

 

[Adrian07]

Thanks for the clearer explanations they basically mirror what I have been thinking, I think the confusion lies in the fact that we have been approaching from different directions. I see time as vertical and distance as horizontal but any forward movement as left to right. I did realize that light seconds were a unit of distance. Thanks for all the communication even if I do not appear to have understood at times it has helped me think.
I now believe that I have worked out why the formulas take the form they do, in fact I now know where the answers to the 3 paradoxes of relativity lay, that is the pole and barn, the twins and the grandfather paradox. The answers to these show where relativity is not completely understood, there are no paradoxes.
For the record T = infinity and distance (length) = 0 do make sense but not within the structure of the universe they actually tell of infinite energy in perfect symmetry with a perfect vacuum, in other words what existed before the BB, although I have no doubt you will disagree.

 

[ttakacs]

Hello to everyone,

I have a question addressing both special relativity and quantum mechanics. On one hand from special relativity we know that the longitudinal length of a moving object decreases with increasing speed, as much as goes to zero with getting close to c.

However on the other hand, at the opposite order of the scale we have Planck length - a very very small value but definitely greater than 0 meter. If this is an absolute lower limit of length, than introducing it to the relativity formula we get a limit speed slightly less than the speed of light, but a new theoretical limit for velocity.

Maybe the main issue whether Planck length is a real lowest limit or just a limit for our observation and there are smaller distances nevertheless cannot be experienced.

Please give me an explanation or resolve this apparent paradox. Thank you.

 

[PeterDonis]

If this is an absolute lower limit of length, than introducing it to the relativity formula we get a limit speed slightly less than the speed of light, but a new theoretical limit for velocity.

The Planck length is a quantum phenomenon, plus it involves gravity; so it isn't covered by special relativity for two reasons, so to speak.

As far as how the transformation laws between frames would have to change if the Planck length were an absolute lower limit on length, just changing the "speed limit" to slightly less than that of light would not be enough by itself, because that "limit" would not be frame-invariant. There is a theoretical proposal called Doubly Special Relativity which modifies the Lorentz transformations at short distances so that the Planck length is frame-invariant (i.e., an object that looks 1 Planck length long in one frame looks 1 Planck length long in all frames).

Another alternative would be to simply say that ordinary relativity stops being valid at short enough distance scales, and some other physics (such as string theory) takes over. This would have to apply to both special and general relativity, since, a I noted above, the Planck length involves gravity.

We can't do experiments at anything like the distance scales we would need to to start testing these alternatives; the Planck length is something like 20 orders of magnitude smaller than the smallest scales we can probe experimentally at this time.

 

[ttakacs]

Thank you. So the problem was something like to indicate time dilation in a very fast inertial spaceship by a mathematical pendulum - without gravity.. I will check the page with double relativity you suggested, thx.

By the way IF Planck length was the smallest length and Planck time was the smallest time could we say that our descriptions of nature with differential equations are not correct (even if very good approximations) since we cannot make limit values in derivatives e.g. dt and dx, etc. cannot go to zero?

Also I have a question related to quantification. In physics continuous fields are often used to describe nature. However mathematics might teach us to be careful with infinity: maybe the best example is the Banach-Tarski paradox that tells indirectly something that physical matter cannot be continuous since it could go against conservation of mass. Could you explain please how this is bypassed in physics?