Speed of light while chasing after it

[Sundu]

Hi. I am new to the forums and I am hoping to learn a few things.

I have been interested in cosmology and its theories for a while, but I have only recently picked up a book on it. It is Brian Greene's The Fabric of the Cosmos. In the third chapter, Greene gives an example of how the speed of light is a constant 670 million MPH, no matter if one is going towards it or moving away from it. In his example, Bart Simpson and Lisa Simpson are conducting an experiment in which Bart uses a 500 million MPH skateboard to chase after a beam of light. While Bart is chasing the beam, Lisa is observing the light to be speeding away from Bart at 170 million MPH (since Bart is going at a constant 500 million MPH; 670 million MPH - 500 million MPH = 170 million MPH). When Bart returns from his trip, he tells Lisa that the light was speeding away at a constant 670 million MPH, not 170 million, while from Lisa's viewpoint, the light was speeding away at 170 million MPH.

As far as I understand it, Greene explains this paradox in this way: "...the input that [Bart] uses to figure out how fast the light is receding from him, are different from Lisa's measurements... experimenters who are moving relative to each other, like Bart and Lisa, will not find indentical values for measurements of distances and durations. The puzzling experimental data on the speed of light can be explained only if their perceptions of space and time are different."

It seems as if the explanation that he gives explains the next concept about the two different motions - motion through space and motion through time, where the faster you are traveling the more energy you are diverting towards movement through space instead of time, thus going through time more slowly from the perspective of another person who is stationary.

But how does this relate to the paradox about the light traveling at a constant speed of 670 million MPH in relation to a traveler who is speeding at 500 million MPH? I still do not understand why this happens; I am going through the text, and I still don't get it for the second day of reading it through. How and why does this 670 million MPH number remain constant in relation to a traveller going at 500 million MPH?

Thank you for your help!

 

[FunkyDwarf]

It is indeed a tricky one and something i grapple with often, and have found a few ad-hoc ways to rationalise it.

If you consider the photon, by definition is massless. According to Newtons laws the only thing (ignoring physical obsticles which DO affect light) that, afaik, can provide resistance to acceleration is mass. Since light has none it is free to travel at the 'max speed' limit which just happens to be this number which is purely arbritary but it has to be something right?

Einstein rationalised it using a mixture of Galilean theory and some other stuff, basically saying that all laws of nature and experiements should give the same answer no matter the reference frame (assuming the experiment is in the same reference frame of course).

It was found that the Maxwell's equation, which describe EM waves and also define a speed for them, were not invariant under the Lorentz transformation, ie were not constant in difference reference frames. In order to rectify this you either needed to ditch the idea of special relativity or the idea that there is universal time. Unfortunately at the time they ditched SR and went with aether which the Michaelson Morley experiment showed was wrong. To that end we now know that each intertial reference frame has its own time, which leads to the time dilation bits in SR.

Hope that helps! Like i said id like to know more about this to so please someone else post!

-G

 

[apope]

i think bart would skateboard off the Earth and gravity would pull him into orbit.

 

[Sundu]

What do you mean by "reference frames," FunkyDwarf?

Apope, Lisa might also be dead by the time Bart comes back, but we are going to sacrifice some realism for the sake of the main question.

 

[pervect]

Nature doesn't actually provide us with "why" answers. But experiments (for instance, the Michelson Morely results) do show that the speed of light is constant, even if they don't come with a written explanation of "why". They also demonstrate that there is a maximum limiting velocity for material particles and that relativity gives the correct equations for the momentum and energy of a particle moving at velocities nearly equal to 'c', while Newtonian expressions are not even close.

 

[damgo]

It helped me to think of it like this: we have a very simple, intuitive notion of how space&time should work. In this simple picture velocities just add up, and distances look the same to everybody no matter how they're moving. So we think if Bart's moving at 500 mmph, he'll be catching up to the light and only see it at 170 mmph.

This simple picture is just wrong. It doesn't describe our universe. Space & time work differently. Distances and times look different to people moving in different ways, and velocities don't add up in the usual manner.

 

[Federation 2005]

Why Light Speed Stays The Same

You're asking about how this works in detail. OK, so I'll give you the details.

Consider the following 3 events
(A) Light comes from sun
(B) Light passes around moon
(C) Light reaches Earth
happening around the time of a solar eclipse. In a reference frame attached to the Earth's center with the X axis joining the Earth, moon and sun, the three events have approximate coordinates

(A): (X = 400, T = -401)
(B): (X = 1, T = -1)
(C): (X = 0, T = 0)

with the units being approximately 250000 miles (or 400000 kilometers) for X, and 1.5 seconds for T.

Now take the same 3 events and relate them to a reference that's moving 4/5 light speed from the sun toward the Earth (e.g. a particle coming out of the sun). Set up its x-axis similarly, with its origin (x,t) = (0,0) coinciding with (C). Then the corresponding coordinates are related by
x = (5X + 4T)/3, t = (5T + 4X)/3
and the corresponding coordinates of the events are
(A): (x = 396/3, t = -405/3)
(B): (x = 1/3, t = -1/3)
(C): (x = 0, t = 0)

The T coordinate changes too. That's why it's called "space-TIME", not "space and time".

The temporal logic of Newtonian space and Minkowski space (the latter being the geometry of Special Relativity) are different. In Newtonian space, the past of an event A is the totality of all events at all times up to the time of A; the future is the totality of all events at all times starting immediately after the time of A. The remaining events (those "neither past nor future") together comprise the 3-dimensional space that is the "snapshot" containing event A. The entire universe is layered into a series of snapshots.

A key attribute is that the "neither before nor after" relation is transitive. Therefore, if three events a, b, c are related by
a is neither before nor after b
b is neither before nor after c
then it would follow that
a is neither before nor after c.

Consequently, Newtonian spacetime has such a thing as a "there and now", instead of just a "here and now". The "there and now" of A is the totality of all events neither before nor after A -- the above-mentioned 3-dimensional snapshot.

In equivalent way of describing this structural feature of Newtonian spacetime is that all *finite* speeds are relative, but infinity is an absolute speed, if such were to exist.

In Minkowski space, transitivity is false. The past of event A is a sphere (and its interior) centered on A that shrinks down at light speed to the point at the instant of the A. The future of A is a sphere (and its interior) centered on A that grows at light speed from a point starting at the instant of A. These sphere's themselves are, respectively, the past and future "light cones" of A.

Thus, if event A is on the Earth, there is a 3 second interval on the moon's timeline that is neither before nor after A. This is directly related to the distance the moon is away, which is about 1.5 light seconds.

(Thus, it also follows that the *spatial* geometry of spacetime can be recovered from the temporal logic of the before-after relation, alone!)

If B, C are two such events on the moon, with B occurring after C, then this would be an example where transitivity fails:
A is neither before nor after B
A is neither before nor after C,
but
B is after C.

Instead of the "neither before nor after"'s of an event A forming a 3-D snapshot of the space A is on, here they form a 4-dimensional continuum of all spacetime that lies outside both light cones. This is called the "absolute elsewhere" of event A.

The distinguishing feature, in a similar way, of Minkowski space is that all finite speeds are relative EXCEPT light speed, while the speed of infinity is now relative!

One of the consequences of this is that the laws of forces has to be modified. Gravity in Newtonian physics takes effect by a transfer of momentum across a distance of space at the speed of infinity. If quantum field theory had been retrofitted to Newtonian gravity, the corresponding particles would have been called "synchrons", distringuished by having zero mass, but a non-zero momentum that they carry in the transfer.

In Minkowski space, the "synchron" is replaced by those transfers that take place on the light cone "luxons" (which a photon is an example of). This is the means by which radiation is transferred in space. Those transfers that take place outside the light cone would be called "tachyons". In effect, this is the means by which Coulomb (1/r)^2 forces are mediated (in quantum field theory, they are the tachyonic virtual particles). No fundamental tachyonic particles are known to exist, and information is widely considered to be impossible to transmit tachyonically.

 

[Federation 2005]

A minor fix. Event (A) should be (X = 400, T = -400) not T = -401. Then the transformed coordinates are (A): (x = 400/3, t = -400/3) and (B): (x = 1/3, t = -1/3).

The previous set of coordinates would have described the trajectory of a tachyon. :) Notice by the way that the transformed speed for the (A)->(B) transit is actually *greater*! With the above fixes, the transformed speeds (A)->(B) and (B)->(C) are both c, which is c = 1 in the units I chose.

 

[rbj]

But how does this relate to the paradox about the light traveling at a constant speed of 670 million MPH in relation to a traveler who is speeding at 500 million MPH? I still do not understand why this happens; I am going through the text, and I still don't get it for the second day of reading it through. How and why does this 670 million MPH number remain constant in relation to a traveller going at 500 million MPH?

Nature doesn't actually provide us with "why" answers. But experiments (for instance, the Michelson Morely results) do show that the speed of light is constant, even if they don't come with a written explanation of "why".

but, i think Einstein was thinking "why", even though he knew about the (negative) result of the Michaelson-Morley experiment, i think Einstein's attitude toward it was "as if God had any choice in the matter." Einstein expected, from a pure thought experiment, for c (in vacuo) to be identical for any inertial observer, because there is no reason for one inertial observer to be preferred over any other, and in a vacuum it is meaningless to speak of this vacuum whizzing past you at a speed of 500 million MPH? anyone who is not accelerated in a vacuum has equal claim to consider themselves as "stationary" even if they are moving (at a constant velocity) relative to each other. if they are all "stationary" (as far as themselves are concerned), why should one have different laws of physics (and a different c results in quantitatively different laws of physics) than another inertial observer?

my long-winded attempt to explain this before.

if you accept that they cannot (they have the same c, G, $\hbar$, and $\epsilon_0$ or whatever parameter of free space), then the only way to reconcile both observers (moving relative to each other) observing the same beam of light to be moving at the same speed, in order for that to happen, they have to observe the other's clock as ticking more slowly than their own.

They also demonstrate that there is a maximum limiting velocity for material particles and that relativity gives the correct equations for the momentum and energy of a particle moving at velocities nearly equal to 'c', while Newtonian expressions are not even close.

it's the intrinsic velocity of all things ostensibly instantaneous, whatever action, be it E&M (which has light as waves) or gravity (which, we believe, has gravity waves that travel at the same speed c) or whatever other action (nuclear forces?).

 

[Areid45]

I agree that the easiest way to comprehend this is to think about it in the sense that massless particles (photons) have nothing holding them back from hitting the "speed limit" of the universe.

 

[Federation 2005]

A minor fix. Event (A) should be (X = 400, T = -400) not T = -401. Then the transformed coordinates are (A): (x = 400/3, t = -400/3) and (B): (x = 1/3, t = -1/3).

The previous set of coordinates would have described the trajectory of a tachyon. :) Notice by the way that the transformed speed for the (A)->(B) transit is actually *greater*! With the above fixes, the transformed speeds (A)->(B) and (B)->(C) are both c, which is c = 1 in the units I chose.

Actually, the previous example describes an ordinary time-like trajectory, not that of a tachyon. The tachyon trajectory would come about by taking X = 401 with T = -400, instead of X = 400 with T = -401. Then, plugging in the transformation rule quoted, you'll find the funny behavior described above concerning how the velocity transforms.