Can't Tell If you are shrinking?

[phinds]

I understand completely what you are saying that motion is relative,

No, you clearly do NOT. If I am traveling at .99c relative to Earth and I shoot a bullet at .2c away from me, it is moving away from me at .2c That's what it MEANS for motion to be relative. The fact that I'm moving at .99c is utterly irrelvant to how fast the object I eject moves relative to me.

Relative to someone on Earth, the bullet will travel at something like .9999c, but that is irrelevant to how *I* see it move. MOTION IS RELATIVE.

 

[TheScienceOrca]

No, you clearly do NOT. If I am traveling at .99c relative to Earth and I shoot a bullet at .2c away from me, it is moving away from me at .2c That's what it MEANS for motion to be relative. The fact that I'm moving at .99c is utterly irrelvant to how fast the object I eject moves relative to me.

Relative to someone on Earth, the bullet will travel at something like .9999c, but that is irrelevant to how *I* see it move. MOTION IS RELATIVE.

I guess I don't understand relative motion, I though that if you were going .99c and shot a bullet at .2c it would only go .01c away from you max because nothing can go faster than the speed of light.

If it actually does go .2 from you then Alice would see an object traveling faster than the speed of light right?

 

[phinds]

I guess I don't understand relative motion

That has been clear all along and is what we have been trying to help you with.

I though that if you were going .99c and shot a bullet at .2c it would only go .01c away from you max because nothing can go faster than the speed of light.

No, that's not how Special Relativity works, as has already been explained to you. Nothing can move faster than c in an inertial frame of reference. Google "relative velocity addition" for a full discussion. This is Special Relativity 101

If it actually does go .2 from you then Alice would see an object traveling faster than the speed of light right?

Again, you are not listening to what you are told. As I said, Alice (assuming Alice is in the frame of reference in which Bob is traveling at .99c) will see it moving at something like .9999c

If Alice were in Bob's frame of reference, she would see it moving at .2c

If Alice were in the bullet's frame of reference, she would see it standing still.

 

[TheScienceOrca]

That has been clear all along and is what we have been trying to help you with.No, that's not how Special Relativity works, as has already been explained to you. Nothing can move faster than c in an inertial frame of reference. Google "relative velocity addition" for a full discussion. This is Special Relativity 101
Again, you are not listening to what you are told. As I said, Alice (assuming Alice is in the frame of reference in which Bob is traveling at .99c) will see it moving at something like .9999c

If Alice were in Bob's frame of reference, she would see it moving at .2c

If Alice were in the bullet's frame of reference, she would see it standing still.

Ok that's what I thought would happen.

So if you look back at my experiment it holds true, it doesn't even require Alice.

It allows Bob to tell if he is moving by himself, as long as he can take his initial test at rest on Alices planet.

Do you see what I mean? He'll get the data results of % of shrinkage not an actual value as all is relative. If he is moving faster than when on Alices planet when shooting an object at .9999c it wouldn't shrink as much as it did relative to when it did on Alices planet. Even if Alices planet was moving as well when he took the test.

This is because he can tell he is moving faster than when he was on Alices planet which means he knows he is moving.

Inversely if the cube is bigger than expected he would now know he was traveling slower than when Alices planet which means Alices planet was moving all on.

Thanks for your patience and helping answer these questions, sorry if I don't make much sense.

 

[phinds]

... It allows Bob to tell if he is moving by himself...

I give up.

 

[TheScienceOrca]

I give up.

You agree that no object can move faster than c.

You agree "shortening" is relative to velocity.

So less change in velocity = less relative change in shortening.

You even said it yourself, so I just don't understand.

 

[TheScienceOrca]

If you know relative shrinkage, if traveling at a higher velocity the relative shrinkage will be lower thus he knows he's moving.

 

[PeterDonis]

If you know relative shrinkage, if traveling at a higher velocity the relative shrinkage will be lower thus he knows he's moving.

No, that's not what will happen; at least not if I'm understanding your experiment correctly. Let me check by describing the experiment again, and saying what Bob will observe given my description:

(1) Bob starts out on a planet (Alice's planet or whatever). He takes 4 identical cubes and fires them in 4 different directions at a velocity approaching $c$, taking great care to ensure that, relative to him, all 4 cubes are moving at the same speed (though in different directions). Then he measures their shrinkage. He finds that all 4 cubes shrink by the same amount.

(2) Bob then gets into his rocket ship, along with the 4 identical cubes (which he recovered after phase 1 above), and flies off at a velocity approaching $c$ relative to the planet. (He can verify his velocity relative to the planet by measurements.) Once he is cruising at a steady velocity relative to the planet, he takes the 4 identical cubes and fires them in 4 different directions at a velocity approaching $c$ relative to him (i.e., relative to the ship he is in), taking great care to ensure that, relative to him, all 4 cubes are moving at the same speed (though in different directions). He then measures their shrinkage. He finds, once again, that all 4 cubes shrink by the same amount.

I suspect that you are under the misapprehension that, in phase 2 of the above experiment, the cubes will shrink by different amounts, as measured by Bob; but they won't, given the conditions of the experiment (that in both phases, all 4 cubes move at the same speed relative to Bob, though in different directions).

 

[TheScienceOrca]

No, that's not what will happen; at least not if I'm understanding your experiment correctly. Let me check by describing the experiment again, and saying what Bob will observe given my description:

(1) Bob starts out on a planet (Alice's planet or whatever). He takes 4 identical cubes and fires them in 4 different directions at a velocity approaching $c$, taking great care to ensure that, relative to him, all 4 cubes are moving at the same speed (though in different directions). Then he measures their shrinkage. He finds that all 4 cubes shrink by the same amount.

(2) Bob then gets into his rocket ship, along with the 4 identical cubes (which he recovered after phase 1 above), and flies off at a velocity approaching $c$ relative to the planet. (He can verify his velocity relative to the planet by measurements.) Once he is cruising at a steady velocity relative to the planet, he takes the 4 identical cubes and fires them in 4 different directions at a velocity approaching $c$ relative to him (i.e., relative to the ship he is in), taking great care to ensure that, relative to him, all 4 cubes are moving at the same speed (though in different directions). He then measures their shrinkage. He finds, once again, that all 4 cubes shrink by the same amount.

I suspect that you are under the misapprehension that, in phase 2 of the above experiment, the cubes will shrink by different amounts, as measured by Bob; but they won't, given the conditions of the experiment (that in both phases, all 4 cubes move at the same speed relative to Bob, though in different directions).

Nope you are in right in that situation would notice no change in relative shrinkage.

Bob takes the first measurements when he is at rest on Alices Planet.

From that point on he can tell if he is going faster or slower then when on Alices planet.

 

[TheScienceOrca]

Based on the relativity of the projected cube to the measurement tape that is on the floor of the ship, it follows the laws of relativity and better yet now thinking about it more, he doesn't even need to land on a planet.

Bob can perform the initial experiment anywhere and if he is moving then his results will change when he does the experiment again.Not only can he tell if he is moving he can tell if he is moving faster or slower than when he took his last result.

Doing this if he can theoretically take tests VERY quickly in an imaginary universe, he can tell his movement in 3 axis, he can plot these points and calculate direction, velocity, and distance relative to any other point, and position relative to any point.

Bob can now plot the universe by himself in his ship.

 

[TheScienceOrca]

Bob also doesn't have to travel at the speed of light, he just needs VERY VERY VERY precise measurements.

The bigger change in velocity the more noticeable the relative "shrinkage" so it will be easier to detect.

 

[TheScienceOrca]

Sorry about posting several replies, I am just posting as I think, I love this forum!

 

[PeterDonis]

Nope you are in right in that situation would notice no change in relative shrinkage.

Bob takes the first measurements when he is at rest on Alices Planet.

From that point on he can tell if he is going faster or slower then when on Alices planet.

How? What kind of measurements does he make? I thought I was describing the same experiment you were trying to describe, but if you think your experiment can let Bob "tell if he is going faster or slower", then you must have some different experiment in mind than what I described. Your descriptions up to now aren't enough to tell what that is, so you'll need to describe it again.

(Btw, your subsequent posts don't make things any clearer. You need to be a *lot* more specific about exactly what Bob does and what measurements he makes.)

 

[PeterDonis]

Based on the relativity of the projected cube to the measurement tape that is on the floor of the ship

I don't understand how this makes any difference. Suppose we add a measuring tape to my description in my previous post. In phase 1 (Bob is on the planet), the cubes all move in a certain way relative to the measuring tape, and Bob records all those motions very precisely. Then in phase 2 (Bob is flying at a velocity close to $c$ away from the planet), if Bob starts the cubes out the same way he did in phase 1 (which seems to be what you intend--after all, if he starts the cubes moving differently, of course they're going to move differently), then all their motions relative to the measuring tape in phase 2 will be *exactly* the same as in phase 1. So Bob won't be able to tell the difference.

 

[TheScienceOrca]

How? What kind of measurements does he make? I thought I was describing the same experiment you were trying to describe, but if you think your experiment can let Bob "tell if he is going faster or slower", then you must have some different experiment in mind than what I described. Your descriptions up to now aren't enough to tell what that is, so you'll need to describe it again.

(Btw, your subsequent posts don't make things any clearer. You need to be a *lot* more specific about exactly what Bob does and what measurements he makes.)

I am very sorry I have aspergers, so I sometimes don't communicate that well I will try to explain the whole scenario from scratch to prevent confusion.

So let me first go over a few aspects of relativity to make sure there are no confusions if you disagree with any of these points please specify exactly which aspects of the following statements that you believe are in question. I would very much appreciate that.

Ok the conditions are as follows;

1) No object can travel faster than the speed of light regardless of frame of reference or any, objects simply can not go faster than the speed of light.

2) A mass shrinks in the direction of travel relative to the velocity, as the mass approaches c it will continue to shrink in the direction of motion.

3) So two identical objects traveling in the same direction one at speed of .5c and one at a speed of .2c, the object traveling at .5c will be shorter in the direction of motion relative to the object traveling at .2c. Even that object at .2c would be shorter in the direction of motion than an identical object at .1c.


Now I will explain Bob's ship, bobs ship is at rest on a planet.

Bob's ship is a very large ship with acres of experiment room and extremely precise measurement equipment.


In the center of his experiment room he has a device which can shoot 1cm blocks at close to c in 4 directions 3 all perpendicular to each other and the fourth a vector sum of the 3 other. These blocks are shot across the ships floor.

The floor is made of identical 1cm cubes that are immovable and have 0 friction with the blocks being projected from the machine.

So when the projectile block is not being shot from the machine it is the same size as the blocks on the floor, no matter what speed bobs ship is travelling.


With this knowledge Bob is now ready to perform his base line experiment.

He fires the machine and with his extremely accurate machines measures the size of the projected block relative to the blocks that are on the floor.

Now this is important, the relative shrinkage is also relative to the relative velocity between the block floor and the block projectile.

The faster the projectile is relative to the floor the greater the relative shrinkage.


This way he can find the relative shrinking at this exact velocity, he will not know what that velocity is, but it will come into use later. The only way he can know which speed he is at is if when the block shot does not change in size relative to the blocks on the floor he will know he is traveling at c.

If Bob's ship is now fires his rockets and moves from the planet he will be able to prove that he is moving by doing the following.


Running the same experiment again.

If the 1cm block shrinks to the shortest shrinkage (speed of light) faster than earlier, bob will know he is moving faster than before, this is why;


Lets say the time to the shortest shrinkage the projectile could become on the planet was 1 second. This is because it took 1 second for the 1cm block to get to the speed of light.

If he is traveling faster than before, the time for the block to the reach light speed will be less as the starting velocity was greater. With Bob's precise measurements this decrease in time can be measured, letting bob not only know he is moving, but by how much as well.

Using this powerful ability Bob can now travel the universe and map his travel.

Let me know if there are any questions with any statements I would love to converse this scientifically.

 

[TheScienceOrca]

I don't understand how this makes any difference. Suppose we add a measuring tape to my description in my previous post. In phase 1 (Bob is on the planet), the cubes all move in a certain way relative to the measuring tape, and Bob records all those motions very precisely. Then in phase 2 (Bob is flying at a velocity close to $c$ away from the planet), if Bob starts the cubes out the same way he did in phase 1 (which seems to be what you intend--after all, if he starts the cubes moving differently, of course they're going to move differently), then all their motions relative to the measuring tape in phase 2 will be *exactly* the same as in phase 1. So Bob won't be able to tell the difference.

Read my post, yes you are right, but the time it took the shrink changes based on v1.

time from v1 to v2 is shorter if v1 is closer to v2 as nothing can accelerate instantly (teleport)

 

[Orodruin]

As you have been told repeatedly, your experiment will not work the way you believe. Yet you continue to argue rather than trying to understand which part of the theory you do not understand. This is not scientific discussion, this is us explaining things to someone who does not want to listen and that is tedious and frustrating. I am done with this thread.

 

[Oldtheorist]

I'm going to reread this when it's not 1 am local but I believe you are not considering time dilation along with length contraction. Time dilation will allow you who are traveling at .9c to launch a projectile that's also traveling at .9c RELATIVE TO YOURSELF and in the same direction of travel, but the projectile itself will never exceed c - not when measured by you or by an outside observer.

 

[Chronos]

I'm not sure what you're saying here. The reduction in distance between the two rockets is happening as though one of them is traveling at 1.8c relative to the other, yes? Why would an external observer not conclude that?

The clocks will disagree.

 

[Orodruin]

The clocks will disagree.

Careful here, he is arguing that an external observer will see the distance between the objects shrink at speed 1.8c, which is true in the frame of the external observer. In this frame there is only one time, the time coordinate of the frame. Then there is a terminology issue in calling this "relative velocity" since this is typically reserved for how fast something moves relative to a given observer.

 

[TheScienceOrca]

I'm going to reread this when it's not 1 am local but I believe you are not considering time dilation along with length contraction. Time dilation will allow you who are traveling at .9c to launch a projectile that's also traveling at .9c RELATIVE TO YOURSELF and in the same direction of travel, but the projectile itself will never exceed c - not when measured by you or by an outside observer.

Right because the time of the universe relative to the projectile is now traveling VERY slowly.

Will this result in even more shrinkage of the projectile or since it is never over c, it won't shrink more.

 

[pbuk]

I had chosen not to join in on this thread TheScienceOrca: you don't explain what you are thinking very well, you don't seem to listen effectively to what other people are saying and you sometimes come over as arrogant.

I am very sorry I have aspergers

I now understand why your posts come over this way: don't worry, lots of other people on this forum also have Asperger's syndrome and nearly everyone in Physics has experience (admittedly not always successful) of working with people with AS.

So I'll see if I can help...

1) No object can travel faster than the speed of light regardless of frame of reference

Let's get away from the words "frame of reference". I don't think you have the maths skills to work with this concept and it is not necessary to an understanding of Special Relativity.

So I'll restate this as: Let c be the speed of light in a vacuum. No object or information can be measured by any observer to be traveling faster than c, and no object with mass can be measured by any observer to be traveling at c.

objects simply can not go faster than the speed of light.

This statement does not make sense: the whole point of SR is that measurements of velocity are relative to the observer, an object does not "simply go" at any speed.

A mass shrinks in the direction of travel relative to the velocity, as the mass approaches c it will continue to shrink in the direction of motion.

This is not correct. Objects moving away from an observer at relativistic speed are seen by that observer to contract in the direction of motion. No change is observed by the object itself.

So two identical objects traveling in the same direction one at speed of .5c and one at a speed of .2c, the object traveling at .5c will be shorter in the direction of motion relative to the object traveling at .2c. Even that object at .2c would be shorter in the direction of motion than an identical object at .1c.

No, the lengths of all the objects remain the same. Each object will measure its own length to be the same. Each object will measure the other objects' lengths as different, the amount of the difference and whether it is an apparent enlargement or contraction depends on how they take the measurement and what they measure their relative velocities to be.

In the center of his experiment room he has a device which can shoot 1cm blocks at close to c in 4 directions 3 all perpendicular to each other and the fourth a vector sum of the 3 other. These blocks are shot across the ships floor.

That doesn't work: if the four blocks are traveling in the directions you describe then a maximum of two of them can be shot across the floor.

I suggest that instead Bob wants to fire 4 blocks at the same speed (measured by Bob) across the (frictionless) floor at 0, 90, 180 and 270 degrees to the direction of travel away from Alice.

Now this is important, the relative shrinkage is also relative to the relative velocity between the block floor and the block projectile.

No, the relative shrinkage (measured by Bob) is ONLY relative to the relative velocity between the floor (and therefore Bob) and the blocks. As Bob fired all the blocks away from him at the same speed from his point of view he measures their shrinkage to be the same.

However because Alice measures the blocks to be traveling at different speeds, she measures their shrinkage as correspondingly different.

This way he can find the relative shrinking at this exact velocity, he will not know what that velocity is, but it will come into use later. The only way he can know which speed he is at is if when the block shot does not change in size relative to the blocks on the floor he will know he is traveling at c.

If Bob's ship is now fires his rockets and moves from the planet he will be able to prove that he is moving by doing the following.


Running the same experiment again.

If the 1cm block shrinks to the shortest shrinkage (speed of light) faster than earlier, bob will know he is moving faster than before, this is why;

Lets say the time to the shortest shrinkage the projectile could become on the planet was 1 second. This is because it took 1 second for the 1cm block to get to the speed of light.

Two problems with this:
1. The block can never get to the speed of light.
2. Bob fires the blocks at a certain speed v across a frictionless surface: their velocity does not change over time.

 

[pbuk]

You should also be aware that there are two different apparent length contraction effects. I won't go into details, I'll just mention that

1. If you measure length by recording the time the two ends of a relativistic object are observed to pass by stationary synchronized clocks you will always measure a length contraction (the Lorentz contraction).
2. If you measure length by taking a photograph of a relativistic object passing a stationary ruler you will measure a contraction if the object is traveling away from you, an expansion if the object is traveling towards you, and a weird distortion if the object is traveling past you. This is the Penrose-Terrell effect (it also goes by other names).

Because of this we need to agree on what we mean when we say "observe", "measure" or "see" the length of an object moving at relativistic velocity.

 

[TheScienceOrca]

Thank you for the in depth reply, I feel like I have a better grasp on relativity. Due to the time dilation objects can move at faster than the speed of light relative to other objects as the time simply changes for the universe relative to the object?

Let me know if this is right, and thanks for the explanation, that is exactly what I wanted to hear and now I know why I am wrong.

 

[TheScienceOrca]

You should also be aware that there are two different apparent length contraction effects. I won't go into details, I'll just mention that

1. If you measure length by recording the time the two ends of a relativistic object are observed to pass by stationary synchronized clocks you will always measure a length contraction (the Lorentz contraction).
2. If you measure length by taking a photograph of a relativistic object passing a stationary ruler you will measure a contraction if the object is traveling away from you, an expansion if the object is traveling towards you, and a weird distortion if the object is traveling past you. This is the Penrose-Terrell effect (it also goes by other names).

Because of this we need to agree on what we mean when we say "observe", "measure" or "see" the length of an object moving at relativistic velocity.

Ah I see, well if it is true to the question I asked above there would be no need to rewrite the concept as it would be wrong.

The whole concept was based off objects not being able to go as fast relative to bob as they would when he is not moving as they would be limited by the speed of light.

 

[pbuk]

the time simply changes for the universe relative to the object?

I think I would rather say that "time is different for other observers in the universe which are moving relative to the object", but you seem to be on the right track now.

There must be a good non-mathematical primer for SR on the web (with animations of passing trains etc), can anyone recommend one to help TheScienceOrca reinforce correct understanding?

 

[phinds]

Careful here, he is arguing that an external observer will see the distance between the objects shrink at speed 1.8c, which is true in the frame of the external observer. In this frame there is only one time, the time coordinate of the frame. Then there is a terminology issue in calling this "relative velocity" since this is typically reserved for how fast something moves relative to a given observer.

Exactly. I'm speaking of "closing speed" which does not need any transformations to bring it down below c. It IS greater than c in the frame of an external observer. This has been discussed many times on this forum and I was surprised to see Chronos apparently contradicting it.

 

[PeterDonis]

Ok the conditions are as follows

MrAnchovy's comments on these look useful to me. I don't really have anything to add here; I think it's more fruitful just to directly comment on your experiment.

He fires the machine and with his extremely accurate machines measures the size of the projected block relative to the blocks that are on the floor.

Now this is important, the relative shrinkage is also relative to the relative velocity between the block floor and the block projectile.

The faster the projectile is relative to the floor the greater the relative shrinkage.

Ok so far.

Running the same experiment again.

If the 1cm block shrinks to the shortest shrinkage (speed of light) faster than earlier

Which won't happen; all 4 blocks will behave exactly the same, according to all Bob's measurements, as they did in the baseline run of the experiment. This is what you don't seem to get; your proposed experiment will not give the results you claim it will.

 

[PeterDonis]

Read my post, yes you are right, but the time it took the shrink changes based on v1.

No, it doesn't.

time from v1 to v2 is shorter if v1 is closer to v2 as nothing can accelerate instantly (teleport)

This is correct as a matter of logic, but it is based on an incorrect premise: you are assuming that "v1 is closer to v2" during the second run of the experiment. It isn't; the "distance" between v1 and v2 is the same during both runs, because of the way velocities transform between frames in relativity. This is what people have been trying to explain to you.

 

[PeterDonis]

The whole concept was based off objects not being able to go as fast relative to bob as they would when he is not moving as they would be limited by the speed of light.

And this is not correct; the speed of light does not "limit" velocities in this sense. You are assuming that velocities add linearly; in SR, they don't. If you want something related to velocity that does add linearly, you need to use the "velocity parameter" (also called the "rapidity") $\alpha$, which is defined by $v = \tanh \alpha$, where $v$ is the ordinary velocity (note that I'm using units in which $c = 1$) and $\tanh$ is the hyperbolic tangent function; $\alpha$ is what adds linearly the way ordinary velocities do in non-relativistic mechanics.

If you look up the $\tanh$ function, you will see that as $v \rightarrow 1$, $\alpha \rightarrow \infty$. So the speed of light corresponds to an infinite velocity parameter $\alpha$, and any finite number is the same "distance" from infinity, so the speed of light does not limit velocities in relativity the way you are thinking it does.

 

[TheScienceOrca]

I think I still had a Newtonian view of everything so it really messed me up in relativity, but now I think I can get it!

 

[TheScienceOrca]

I didn't know that the shrinkage was based on relativity and that objects can move in reference to you at speeds over c even though they are not traveling over c.

It makes complete sense now.

 

[PeterDonis]

objects can move in reference to you at speeds over c even though they are not traveling over c.

This may be a misunderstanding on your part, or it may just be bad phrasing. Normally, an object moving "in reference to you" means the velocity that you measure the object to have, relative to you. That can never be greater than $c$ (and it can only be $c$ for light or other massless objects, not for any object with nonzero invariant mass).

What can be greater than $c$ is the "closure speed" between two objects, both moving relative to you, as measured by you. But if, for example, object A is moving at $0.8c$ relative to you, and object B is moving at $0.8c$ relative to you in the opposite direction, then, while the closure speed between A and B, relative to you, is $1.6c$, the speed of B relative to A (meaning, the speed A measures B to have in reference to A), or of A relative to B (meaning, the speed B measures A to have relative to B), is *not*; it is only $0.976c$ (approximately).

Once again, this is because of the way velocities add in relativity. One way of seeing it is to use the fact, which I mentioned in a previous post, that $v = \tanh \alpha$, and that $\alpha$ adds linearly. So if we have $\alpha_A$ and $\alpha_B$ as the velocity parameters of A and B, relative to you, then the velocity parameter of A relative to B (or B relative to A) is $\alpha_{AB} = \alpha_A + \alpha_B$, and using the rule for adding hyperbolic tangents gives us (again, I'm using units where $c = 1$):

$$
v_{AB} = \frac{v_A + v_B}{1 + v_A v_B}
$$

If you plug in $v_A = v_B = 0.8$, you will find $v_{AB} \approx 0.976$, as I said.

 

[TheScienceOrca]

What would Alice see standing on a planet if a ship going .9c flies by parallel to her and shoots a bullet in the same direction of travel .9c.

You earlier stated that the cube would fly relative to the fast moving ship and not be limited by the speed of light.

So would Alice now see a bullet going 1.8c and a rocket going .9c?

I am uncertain on a few question still sorry

 

[Nugatory]

What would Alice see standing on a planet if a ship going .9c relative to Alice flies by parallel to her and shoots a bullet with a muzzle velocity of .9c relative to the ship in the same direction of travel .9c.

(I've tried to guess what you're asking and have clarified accordingly. You really have to get in the habit of never stating a velocity or speed without also considering what it is relative to)

If I have guessed correctly what you're asking, Alice will see the ship moving at a speed of .9c relative to her, and will see the bullet moving at a speed of .994c relative to her.

I got this result from the relativistic velocity addition formula: $w=\frac{u+v}{1+uv}$ (and measuring distance in light-seconds and time in seconds so that $c=1$ and don't need to clutter things up with factors of $c$ and $c^2$)