The message only needs to be encrypted for transmission over an insecure channel (SMTP). Google could store all your messages in plaintext for searching purposes, and only encrypt and decrypt when sending or receiving a message.
When I say p approaches infinity as v approaches c, I am saying there exists no limit of the function p(v) as v approaches c. The mathematical definition of this is basically that no matter how high a number you pick for p, I can always find a value of v which is less than c that produces it. In other words, the bigger your momentum gets, the closer your velocity gets to c.
An important aspect in relativity is that you have to take into account the observer. Relativity says a massive object cannot be travelling at the speed of light in any reference frame.
In the reference frame of an outside observer, your objects a and b are each travelling at half the speed of light. You are correct in saying their relative velocity is c, based on how relative velocity is defined (simply the addition of two velocities) but neither of the objects is travelling faster than the speed of light.
To find out the speed of one object in the reference frame of the other object, we must use the special relativity formula for the addition of two velocities u and v:
(u + v) / (1 + uv/c^2)
If obejcts a and b are spaceships, an astronaut in spaceship a would see spaceship b travelling away from him at 0.8c, obviously not in excess of the speed of light. For a good discussion of this, see this site.
As to your second question, obviously the speed of light is not infinite speed. However, a massive object travelling at the speed of light would have to have infinite momentum. The relativistic formula for momentum is:
p = mv / sqrt (1 - v^2/c^2)
as v approaches c, p approaches infinity. It would require an infinite amount of force to accelerate anything to a state of infinite momentum (or a finite amount of force applied over an infinite time). Since neither of these things are possible, everything with mass must travel at less than the speed of light.
Also note that by definition an object's momentum cannot be greater than infinity, so just by the limit of the momentum formula we can deducde that velocities greater than the speed of light are impossible.
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The message only needs to be encrypted for transmission over an insecure channel (SMTP). Google could store all your messages in plaintext for searching purposes, and only encrypt and decrypt when sending or receiving a message.
When I say p approaches infinity as v approaches c, I am saying there exists no limit of the function p(v) as v approaches c. The mathematical definition of this is basically that no matter how high a number you pick for p, I can always find a value of v which is less than c that produces it. In other words, the bigger your momentum gets, the closer your velocity gets to c.
In the reference frame of an outside observer, your objects a and b are each travelling at half the speed of light. You are correct in saying their relative velocity is c, based on how relative velocity is defined (simply the addition of two velocities) but neither of the objects is travelling faster than the speed of light.
To find out the speed of one object in the reference frame of the other object, we must use the special relativity formula for the addition of two velocities u and v:
(u + v) / (1 + uv/c^2)
If obejcts a and b are spaceships, an astronaut in spaceship a would see spaceship b travelling away from him at 0.8c, obviously not in excess of the speed of light. For a good discussion of this, see this site.
As to your second question, obviously the speed of light is not infinite speed. However, a massive object travelling at the speed of light would have to have infinite momentum. The relativistic formula for momentum is:
p = mv / sqrt (1 - v^2/c^2)
as v approaches c, p approaches infinity. It would require an infinite amount of force to accelerate anything to a state of infinite momentum (or a finite amount of force applied over an infinite time). Since neither of these things are possible, everything with mass must travel at less than the speed of light.
Also note that by definition an object's momentum cannot be greater than infinity, so just by the limit of the momentum formula we can deducde that velocities greater than the speed of light are impossible.