When a mathematician says 'imaginary', he does not mean the same thing as when other people says it. To many of us, 'imaginary' means 'something in the head'. There is no difference, for instance, between a 'real' chair and an 'imaginary' one, in terms of how they can look. As we will see, it means different thing in maths. Unfortunately, physicists sometimes uses 'imaginary' in a confusing manner. They use the word to refer to the mathematical 'imaginary', but without emphasizing that it is the mathematical 'imaginary', and not 'imaginary' in the physics sense. An example of how they use 'imaginary' this way is when they say that 'time is along an imaginary axis in Minkowski spacetime'. Someone might easily think that the quality of fourth dimension being unobservable in 3d is what makes time 'imaginary' in this case. In reality, it is nothing of the sort!
The other thing I will show you is how physicists walk away with actually being illogical by invoking imaginary numbers as though they can be used in physical quantities. Does 'imaginary distance' have any meaning in physics, with 'imaginary' in the mathematical sense? Or I can ask: 'can physical quantities assume imaginary numbers'?
In a nut shell, 'imaginary number' is just a number multiplied by the square root of negative 1, written as 'i'.
A 'complex number' on the othet hand, is just a real number plus an imaginary number. So again 'complex' means different thing in mathematics. In the usual speech, a 'complex' thing has myriads of parts. But in math, 'complex number' is realy not a complex number in this way. It just has two parts: a+bi.
Complex numbers arises when solving quadratic equations. Here is how they are solved:
The solution for the quadratic equation is a complex number simply if the value under the squaroot is less than zero, ie is a negative number.
Now let us see how they ought to be interpreted in physics. We will consider an object thrown upwards, under gravity. The velocity,v, of such an object, reduces steadily until it reaches zero, then in 'turns negative', meaning it now moves downwards. So the height, h, of the object increases until its maximum, hmax, and then it begines to reduce. Also note that v is given by the rate of change in h with time, ie the 'derivative of h'. So h is the 'antiderivative of v', which is same as 'integration of v with respect to time'. The result is that h is a quadratic equation with t as the variable.
That is for accelerating g. For deceleration one, use -g. Now to get the maximum height, we use the value for 't' when v=0. This is because at the maximum height, the object stops. We will also solve the equation for t, or 'write t as the subject of the formular'. So we will use the earlier formular for the solution of a quadratic equation.
This is the solution for 't'. Now to check at what time the height is at its maximum:
We are now ready to interpret complex numbers as applied in the real world. Note that at the maximun height, the value under the square root becomes zero. Any height above that, the value under the square root becomes anegative number! So complex number has a straight foward interpretation: There is no time when the value under the square root is negative! This is because at no time does the height of the ball exceeds u/2g. To pretend that time can have an imaginary value is to say that a stone goes higher than it actually does!! It is simply being illogical and contradicting the facts!!
Since that is the case, how comes physicists tells us that physical quantities can attain imaginary values? Here are 4 examples where physicists makes absolutely nonsensical claims by not paying attention to the contradiction that comes by thinking that complex numbers can describe real physical quantities:
1.) NEGATIVE ENERY
As seen in the kinetic energy formular: E=1/2mv^2, if E becomes negative, then v becomes a squareroot of a negative number. So when interpreted consistently, there can be no velocity that can give a 'negative energy'. So what happens at the conversion of the supposed 'negative energy' into kinetic energy?
2.)BLACK HOLES
Consider the formula for the 'gravitational time dillation':
As you can see, there is a point beyond which there is no time dilation! This is precisely at 2GR=c^2, which is the Schwarzchild Radius. What the equation tells us is that like a ston thrown upwards cannot reach more than heigh u/2g, no object can move further than the schwarzchild radius, or further than the event horizon! This means that blackholes cannot form, nor can anything enter the event horizon. The equation actually negates the reality of black holes! To say that 'black holes can form', and ignore the meaning of 'imaginary time dilation' is as logical as saying that a stone thrown slightly upwards can actually reach the moon just because one can express the 'time' for such an event as a complex number! What nonsense!!
3.WAVE FUNCTIONS
The solutions for quantum particles are all given in complex numbers! This means that the describe quantities can cannot be attained in nature! Should real things be described by real numbers?
4.TIME AS FOURTH DIMENSION
As we saw in previous blog post, Lorentz Transform can only be a 'rotation in a four d space time' if time is imaginary. So as we have seen, this means that we can only describe Lorentz transforms as 'rotations in a 4d spacetime' if there is no time when such things happen!! In other words, what we actually have is a REDUCTIO-AD ABSURDUM against the idea that 'we are living in a 4 d space time'! However, by foolishly failing to pay attention to the meaning of complex numbers, the physicists ended up turning it on its own head!!
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