# Invariance and relativity

Nugatory
Mentor
So using my previous example please explain using the math why the object that bounces off of the wall with enough momentum in Bob's frame to kill him but either does not bounce off the wall in Alice's frame or bounces off the wall with less momentum than required to kill Bob in Alice's frame, explain please how these phenomenon all end up with the same outcome.
Your triggering device is responding to the magnitude of the four-momentum, which will be the same no matter which frame you use to calculate it.

You can't build a device that triggers off of the value of a frame-dependent quantity such as the three-momentum, for the same reason that you can't build a device that will trigger or not according to whether the person watching (that's "watching"! - not "interacting with"!) the device is moving or at rest relative to the device.

Thanks.
For those of us that don't understand 3 momentum vs 4 momentum can you give a quick example?

Thanks.
For those of us that don't understand 3 momentum vs 4 momentum can you give a quick example?
This is not the example you want but you should remember that the relative velocity between the ball and Bob's head is frame invariant.
Everyone sees the ball strike with the same force.

Janus
Staff Emeritus
Gold Member
Thanks.
For those of us that don't understand 3 momentum vs 4 momentum can you give a quick example?
So using my previous example please explain using the math why the object that bounces off of the wall with enough momentum in Bob's frame to kill him but either does not bounce off the wall in Alice's frame or bounces off the wall with less momentum than required to kill Bob in Alice's frame, explain please how these phenomenon all end up with the same outcome.
But it will have enough momentum to kill Bob is both frames. That's the whole point behind the Lorentz transforms. You start with the fact that anything that happens according to one frame happens according to all frames. Then you consider that c is invariant. The Lorentz transforms are what allows any frame maintain the consistency of both these facts. In other words, in order for Alice to agree with Bob that the object kills him and measure the speed of light relative to herself as being c, she has to measure Bob and his ship as length contracted, time running slow for Bob, and a different clock synchronization than Bob does. The Lorentz transforms are what maintain this consistency of events and the invariance of the speed of light.

Let's take this simple example. IN Bob's frame, we have two objects of 1kg each moving at 0.01c relative to the ship and towards each other until they hit in an non-elastic collision (like two balls of clay) and stick together. The resulting 2kg mass will be motionless with respect to the ship and have 0 momentum as measured by Bob.

Now consider what Alice would conclude. For her the ship (and Bob) are moving at 0.99c. Using the addition of velocities formula it works out that the velocity of one object will be 0.990197c and that of the other object will be 0.989799c. This gives them speeds of 0.000197c and 0.000201c with respect to the ship according to Alice. At first blush it then seems that they could not possibly collide, stick together and have the resulting mass remain motionless with respect to the Ship.

However, this first impression is wrong. Let's work out the whole problem from Alice's frame:

One object with a rest mass of 1kg is moving at 0.989799c, its momentum is $\rho = mv\gamma$ where $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ and m is the rest mass for the object
For ease of calculation, we will use units where c=1, which gives us a numerical value for its momentum of 6.94738.

The other object is moving at 0.990197c and has a momentum of 7.08915( in the same direction). So after the objects collide and stick, the resulting mass will have a momentum of 14.03653. Plugging this and a rest mass value of 2 kg into the momentum formula above gives the resultant velocity of this mass with respect to Alice. This works out to 0.99c, or the same speed as Bob and his ship. In other words, both Bob and Alice agree that according to their own measurements and observations, that the result of the collision is a mass that remains motionless with respect to the ship.

The fact that the two masses had different speeds with respect to Bob as measured from Alice's frame did not, in the end, change the result as far as Alice was concerned. Assuming that it did would lead to a false conclusion. The same type of false conclusion you are making when you assume that an object thrown against a wall and coming back and hitting Bob hard enough to kill Bob in his own frame, would not hit him not hard enough to do so as measured from Alice's frame.

Thanks a lot for the very detailed explanation. I learned a lot from this thread.