## Sunday, 26 June 2016

### Breaking the ultimate speed of c

(moved to: http://niralaashutosh.blogspot.in/2016/06/breaking-ultimate-speed-of-c.html)
DISCLAIMER: I am not a physicist, nor do I understand special theory of relativity properly, so I am sure that my logic is flawed, but if you could pin-point the flaw in simple terms, please leave that in the comment or mail me.

As I understand: an object with finite mass cannot ever touch speed of light because as its speed increases it's mass also increases as m0/sqrt(1- (v/c)^2). Thus mass approaches infinity as speed approaches c, and infinite force can not be created to accelerate the object with infinite mass.

The Idea: What would happen if the force accelerating them is generated by the same objects' mass? Consider two massive objects approaching each other. If conditions are generated such that they approach near c, well before collision, then: their mass would start increasing but so would the gravitational pull between them. This should even cause further acceleration, and thus it may be possible, with right settings, to accelerate the object to speed of light (and possibly beyond!!!) before they collide.

Now, consider the setting, where two massive objects with rest mass of m0 and separated by distance ro, are at rest. Now, suppose a force of k*m0 is applied on both the objects so that they start accelerating towards each other.

Also there is gravitational pull acting on them as shown below:

Now as both the objects start moving towards each other, their velocity increases, and so does their mass. Once their velocities approaches c & r reduces, external force k*m0 becomes negligible and it could be ignored, for simplicity.

Gravitational force as experienced by both the objects = Gm^2/r^2
where:
m = m0/sqrt(1-(v/c)^2)
r = current distance

Even after ignoring the effect of k*m0, the above equation is incorrect (as gravity is also bounded by c), but I would come to that later.

So acceleration for both the objects = F/m = Gm/r^2

Few things to note:
• Acceleration is independent of the object's mass
• It is dependent on the other's object mass, and is proportional to it, which is increasing
• Radius is decreasing which further leads to increase in gravitational force, and thus acceleration.

So if these objects have been accelerated to a sufficiently high speed, and if the distance between them is sufficiently large, gravity between them should be sufficient to pull them towards each other, while increasing their speed to c before they collide.

Let's backtrack from the point when the objects would have achieved the speed c. Suppose the objects attain the speed c when they are separated by a distance d.
We know that del V = a * del t
For simplicity let us consider del t = 1 sec (before achieving c), and let us consider a constant acceleration during this period. We know that it would be increasing during this period, but we will consider its least value during this period, which would be the one at the beginning of this period.

1 sec ago, if the distance between the objects was r then acceleration at that time a = Gm/r^2So v = c-Gm/r^2 (as v=u+at for uniformly accelerated motion and
t=1s in this case)

We can assume any values of v and r. Let us assume
v = 0.99c and
r = 2c

For these values of v and r, let us find which value of m satisfies the above equation
=> m = ?
v= c - Gm/r^2
=> Gm/r^2 = c - v
=> m = (c - v)*r^2/G
=> m = (c - 0.99c) *r^2/G = 0.01c*r^2/G
=> m = 0.01c*4c^2/G             (as r^2 = 4c^2)

G = 6.67*10^(-11)  = 2.23*10^-19 c
=> m = 0.01c *4*c^2/(2.23^-19 c) = 0.018c^3 = 0.4888 * 10^24 Kgs

Please note that this is the mass of the object when it's speed is 0.99c
So rest mass of the object m0 = m*(sqrt(1-0.99^2))= 0.14m (not much difference in terms of order of magnitude)

So if two objects having mass of the order of 10^24, which is of the order of earths mass, are accelerated such that when they are at a distance 2c apart they have a speed of 0.99c, they would cross speed of light well before they collide.!!!!!!

In the above equations I have considered mass of both objects equal, even when they are moving. Since gravity also travels at the speed of light one object would always feel the other object lighter at each moment! (Would have tried to derive the exact equation but Maths mein dabba gul. Would be grateful if someone do it for me)

But even with the above correction, the acceleration would keep on increasing due to gravity and the objects would break the c barrier, of the ultimate speed of light, before collision.

What would be the repercussions if the above is true? What would happen the moment the objects achieve c?
Their mass would become infinite, causing gravity to become infinite, thus sucking everything in the universe in itself with infinite speed (effect would propagate with speed of light). And the universe would be destroyed!

Of course, this can't happen because of conservation of energy. To start with, the system of two objects does not have enough energy to suck entire universe, so no matter what, they can't do it!! (Perpetual Motion machines can't be created, no matter how elegantly we design the system because fundamental thing is: "Energy is always conserved")

My questions:

• Where is the flaw?
• Probably in calculating gravitational force?
• Why does mass of an object increases as it approaches c?
• I suspect the catch should be in the way gravitational force is produced because of mass.