Real and imaginary proper time.
Posted Dec11-08 at 11:12 PM by yuiop
While struggling to make sense of some of the apparently contradictory qualities of physics below the event horizon of a black hole, I made a breakthrough in resolving the issues by considering the relationship of real and imaginary proper time. It turns out that relationship is not actually complex in the usual sense, since physically real motion may have real or imaginary proper time, but complex time with both none zero real and none zero imaginary parts is unphysical.
In the images above, the green and magenta curves are the trajectory of a particle in proper time. The blue and black curves are the trajectory of the same particle in coordinate time. The vertical red line marks the event horizon and the solid vertical time axis on the left marks the centre of the black hole. In the first image only the parts of the curves that have purely real time components are plotted. It can be seen that they are in agreement on the apogee side of the event horizon but below the event horizon the motion of the particle is not defined in terms of pure real coordinate time. In the second image the apogee is below the event horizon, but the motion in terms of real coordinate time and real proper time are in total contradiction.
This contradiction is resolved by allowing imaginary proper time to be physically significant below the event horizon. In the image below, the motion that has an imaginary time component is plotted and it can be seen that the motion in proper time is now in complete agreement with motion in coordinate time on the apogee side of the event horizon.
In the full explanation given in this section of my website it is shown that coordinate motion that has both real and imaginary time components is not physically real. Proper time on the other hand can be real (positive or negative) or imaginary relative to coordinate time. This is the key to resolving many of the apparently paradoxical issues of physics below the event horizon.
The derivation of the equations of motion used in the images is shown here.
Related issues and concepts are discussed in other parts of this blog and at my website.
Any feedback would be welcome.
Thanks.
In the images above, the green and magenta curves are the trajectory of a particle in proper time. The blue and black curves are the trajectory of the same particle in coordinate time. The vertical red line marks the event horizon and the solid vertical time axis on the left marks the centre of the black hole. In the first image only the parts of the curves that have purely real time components are plotted. It can be seen that they are in agreement on the apogee side of the event horizon but below the event horizon the motion of the particle is not defined in terms of pure real coordinate time. In the second image the apogee is below the event horizon, but the motion in terms of real coordinate time and real proper time are in total contradiction.
This contradiction is resolved by allowing imaginary proper time to be physically significant below the event horizon. In the image below, the motion that has an imaginary time component is plotted and it can be seen that the motion in proper time is now in complete agreement with motion in coordinate time on the apogee side of the event horizon.
In the full explanation given in this section of my website it is shown that coordinate motion that has both real and imaginary time components is not physically real. Proper time on the other hand can be real (positive or negative) or imaginary relative to coordinate time. This is the key to resolving many of the apparently paradoxical issues of physics below the event horizon.
The derivation of the equations of motion used in the images is shown here.
Related issues and concepts are discussed in other parts of this blog and at my website.
Any feedback would be welcome.
Thanks.
Total Comments 2
Comments
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"This contradiction is resolved by allowing imaginary proper time to be physically significant below the event horizon. "
Many contradictions can be resolved by using imaginary numbers, but are they truly resolved? Out of curiosity what do you think imaginary proper time is for an observer?Posted Jun10-10 at 08:40 PM by Passionflower 
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The imaginary dimension
It strikes me that while we talk about imaginary numbers, particles etc we are missing something. Complex numbers are fundamental to Maths and Physics.
There can be an "imaginary" dimension - the 5th dimension (after 3 physical and time). There is ample (if indirect) evidence for this. We do not need to tie ourselves up in strings, membrane etc who require multiple dimensions for which there is no evidence. I have had this view for over 20 years and feel the time has come to accept the the universe is "complex", with an "imaginary" dimension - this answers many questions and has strong empirical support.
Ken McCluskeyPosted Oct25-11 at 10:30 AM by ken mccluskey
