Special Relativity: 1st & 2nd Postulates Explained

[rocketboy]

Hello everyone.

I've just received my gr 12 physics textbook today and began reading the chapter on Special Relativity, because I find it fascinating. So I'm new to the topic, please bear with me.

The first postulate states that "the laws of physics have the same form in all inertial reference frames".

The second states that "light propagates through empty space with a definite speed c independent of the speed of the source or observer".

I'm having difficulty connecting the two, and grasping the concept of light having the same definite speed c no matter the reference frame. Don't the laws of physics discussed in the first postulate include those in mechanics about frames of reference?

Say you are moving at 100km/hr down the road and you see a car out your window, and measure that he is doing 20km/hr (from your frame of reference) that you have to add that to your speed, to find out that his speed is actually 120km/hr to the frame of reference of somebody on the side of the road? This means that from the two different reference frames the car is doing two different speeds (relative to each frame of reference).

However, postulate 2 doesn't seem to follow this, because it says that the speed of light is definite, no matter what frame of reference the observer is in. Which in the case of my car example, isn't true.

Could somebody please explain this to me, or show me where I went wrong? I'd like to fully understand this because it's making the section on Simultaneity difficult to understand, and I'm not even going to go onto the Twin Paradox until I have this down.

Thanks.
-Jon

 

[russ_watters]

Those postulates throw a nasty little monkey wrench into that speed calculation you just did, don't they? How that is resolved is that speeds no longer add directly. In order for two observers moving relative to each other to measure the same speed of light, the dimensions of time and space must be different for them. And they are. The simplest example of this is in GPS satellites. Due to their speed, the clocks aboard the satellites lose time when compared to those on the ground (though at the same time, they gain speed due to the reduced gravity).

 

[mikeu]

However, postulate 2 doesn't seem to follow this, because it says that the speed of light is definite, no matter what frame of reference the observer is in. Which in the case of my car example, isn't true.

However, the point of a postulate is that it must be true by assumption (at least so long as you are working within the system it is helping to define). If you take it as true, then it turns out (as you'll learn in your course) that what isn't true is you're assumption that the speed of the other car must be 120km/h.

If you are driving at 100km/h relative to an inertial frame and you observe another car to be driving at 20km/h relative to you, then using the velocity addition rule of special relativity you find out that the other car is in fact going at 119.99999999999999794km/h... Which for all intents and purposes in every day life is 120km/h, which is why our experience tells us that the answer is simply 120km/h.

What's happening here is that in order to maintain the second postulate you listed, the old Galilean velocity addition rule v=u+w has to be abandoned in favour of the Lorentz velocity addition rule,
$$v=\frac{u+w}{1+\frac{uw}{c^2}}$$

You can see that in this equation that if you let the speed of light become infinite then the denominator becomes 1+0=1, so you end up back at the old rule of v=u+w. But since the speed of light is finite, we get that tiny change in the 15th decimal place of the calculated speed above.

 

[rocketboy]

The simplest example of this is in GPS satellites. Due to their speed, the clocks aboard the satellites lose time when compared to those on the ground (though at the same time, they gain speed due to the reduced gravity).

Sorry, how do these clocks work? How do they measure time and how does speed and gravity affect this?

EDIT:

Nevermind, I found that other thread on the gps satelites and followed the link.

Still don't quite understand:

How that is resolved is that speeds no longer add directly. In order for two observers moving relative to each other to measure the same speed of light, the dimensions of time and space must be different for them. And they are.

 

[russ_watters]

Sorry, how do these clocks work? How do they measure time and how does speed and gravity affect this?

By energizing a cloud of cesium gas and measuring the frequency of light they give off: http://tycho.usno.navy.mil/cesium.html

Still don't quite understand:

You asked how speed and gravity affect how clocks work. You're not looking at it quite right: Its tough to accept, but what these clocks show is that speed and gravity affect the rate of the passage of time itself. Or, perhaps more correctly, people in different frames of reference will not necessarily agree on the time lapse between two events or a distance traveled by an object.

http://www.glenbrook.k12.il.us/gbssci/phys/Class/relativity/U7l2b.html is more on the concept, including, if you want it, the derivation of the equation for time dilation.

 

[rocketboy]

Thank you for your help.

Does this mean that the following is true?

Say we built a spaceship that could travel at 1.1c. This spaceship also has headlights. If we were to turn on these headlights while going at top speed, would we pass the waves of light that were emitted by our headlights, since we are traveling at 1.1c and the light emitted is only traveling at 1.0c?

Then say that we had a cannon on the front of the ship, and it shoots tennis balls. Ignoring the fact that nothing can accelerate past the speed of light, if we shot a tennis ball out of the front of the ship, say for simplicity's sake at 0.1c, would the tennis ball be traveling at 1.2c, yet the light emitted from the front of the ship only at 1.0c? (From an outside frame of reference).

I have never delved into such an interesting topic before, I started reading "A Brief History of Time" by Stephen Hawking today. Unbelievable.

-Jon

 

[LeonhardEuler]

Thank you for your help.

Does this mean that the following is true?

Say we built a spaceship that could travel at 1.1c. This spaceship also has headlights. If we were to turn on these headlights while going at top speed, would we pass the waves of light that were emitted by our headlights, since we are traveling at 1.1c and the light emitted is only traveling at 1.0c?

Then say that we had a cannon on the front of the ship, and it shoots tennis balls. Ignoring the fact that nothing can accelerate past the speed of light, if we shot a tennis ball out of the front of the ship, say for simplicity's sake at 0.1c, would the tennis ball be traveling at 1.2c, yet the light emitted from the front of the ship only at 1.0c? (From an outside frame of reference).

I have never delved into such an interesting topic before, I started reading "A Brief History of Time" by Stephen Hawking today. Unbelievable.

-Jon

Actually, there is a problem in assuming that it's possible to build such a ship. You have to remember that things which seem constant like the passage of time and the lengths of distances all depend on the observer in relativity. Suppose you built a spaceship that could travel faster than light relative to the earth. Suppose it was L meters long while it was not moving. How long would it seem to be when traveling at 1.1c? Using some reasoning based on the postulates of relativity one arrives at the conclusion that if an object has length L at rest, the apparent length at speed v is given by
$$L'=L\sqrt{1-\frac{v^2}{c^2}$$
So the apparent length of the spaceship is:
$$L'=L\sqrt{1-1.21} \approx .458iL$$
where i is [itex]\sqrt{-1}[/tex]. In relativity no object can be accelerated up to or past the speed of light. Think about it: the speed of light is supposed to be the same for all observers. When you are in the frame of an object, it does not appear to be moving. If you were in the frame of light, it would appear stationary, but this is impossible because c>0. It is very difficult to get a solid concept for these things. If you really want to understand it, try to derive the Lorenz transformations by yourself from the postulates. Then you will clearly see why all of these bizarre conclusions are necessary. You might want to read farther in your text before you attempt this.

Edit: By the way, suppose the ship fires its cannon and someone on the ship sees the object leave at .5c. Then someone who sees the ship traveling at .9c and sees the projectile fly straight out ahead of the ship would not see the projectile travel at 1.4c., he would see it travel at about .9655c!

 

[jtbell]

Say we built a spaceship that could travel at 1.1c.

You can't, in the context of special relativity. Therefore your following question is meaningless in the context of special relativity.

Someone will probably pop up at this point and mention tachyons, so I'd probably better say something about them. The mathematics of relativity seems to allow for the existence of objects that can travel only at speeds greater than $c$, but never at that speed or less than it. However, to date nobody has found any evidence that tachyons exist, as far as I know. Besides, if you actually want to use a spaceship, it has to start out at rest so you can board it!