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yuiop
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I thought it would be interesting to do a visualisation of Brian Greene's concept that "everything moves at the speed of light". For a preamble see this old thread https://www.physicsforums.com/showthread.php?t=398590
First I should warn that speed or velocity is not defined in the conventional way here. It uses a concept of 4 space dimensions comprising 3 of the normal spatial dimensions and a fourth dimension with units of cTau where Tau is the proper time of a particle. Speed is here defined as dU/dt where U = √(c^2Tau^2+x^2+y^2+z^2) and t is the conventional coordinate time. It is important to notice that this is not the conventional four velocity. (Note that all the signs are the same). For lack of a better term I will call dU/dt the G-velocity and all the dimensions other than the coordinate time collectively as G-space after Brian Greene.
For simplicity I will only consider the x spatial dimension.
For a particle moving relative to the observer, the G-velocity through through G-space is:
[tex] \frac{dU}{dt} = \frac{\sqrt{ (c d\tau)^2+d x^2}}{dt} = c [/tex]
(See the green path in the attached 3D graph for a particle with a 3 velocity of 0.7071c.)
For a particle that is at rest with respect to the observer this reduces to:
[tex] \frac{dU}{dt} = \frac{c d\tau}{dt} = c [/tex]
This is the idea that a stationary particle moves purely through the (proper) time dimension at the speed of light.
(See the blue path in the attached 3D graph)
For a photon the equation reduces to:
[tex] \frac{dU}{dt} = \frac{d x}{dt} = c [/tex]
This is the idea that a light particle moves purely through the spatial dimensions (of 3 space) at the speed of light. Note that there is no component of the photon's path in the proper time dimension.
(See the red path in the attached 3D graph)
The full graph would require plotting in 5 dimensions, (3 spatial + coordinate time + proper time) so only the x spatial dimension is shown along with the two time dimensions in the attached chart. Although it is not very clear from the image, all the paths lie on 45 degree cone, of which a quarter is shown in the image, with the apex at the origin.
In summary, nothing can remain stationary in G space. You are either moving through 3 space or moving through the proper time dimension or a combination of both.
First I should warn that speed or velocity is not defined in the conventional way here. It uses a concept of 4 space dimensions comprising 3 of the normal spatial dimensions and a fourth dimension with units of cTau where Tau is the proper time of a particle. Speed is here defined as dU/dt where U = √(c^2Tau^2+x^2+y^2+z^2) and t is the conventional coordinate time. It is important to notice that this is not the conventional four velocity. (Note that all the signs are the same). For lack of a better term I will call dU/dt the G-velocity and all the dimensions other than the coordinate time collectively as G-space after Brian Greene.
For simplicity I will only consider the x spatial dimension.
For a particle moving relative to the observer, the G-velocity through through G-space is:
[tex] \frac{dU}{dt} = \frac{\sqrt{ (c d\tau)^2+d x^2}}{dt} = c [/tex]
(See the green path in the attached 3D graph for a particle with a 3 velocity of 0.7071c.)
For a particle that is at rest with respect to the observer this reduces to:
[tex] \frac{dU}{dt} = \frac{c d\tau}{dt} = c [/tex]
This is the idea that a stationary particle moves purely through the (proper) time dimension at the speed of light.
(See the blue path in the attached 3D graph)
For a photon the equation reduces to:
[tex] \frac{dU}{dt} = \frac{d x}{dt} = c [/tex]
This is the idea that a light particle moves purely through the spatial dimensions (of 3 space) at the speed of light. Note that there is no component of the photon's path in the proper time dimension.
(See the red path in the attached 3D graph)
The full graph would require plotting in 5 dimensions, (3 spatial + coordinate time + proper time) so only the x spatial dimension is shown along with the two time dimensions in the attached chart. Although it is not very clear from the image, all the paths lie on 45 degree cone, of which a quarter is shown in the image, with the apex at the origin.
In summary, nothing can remain stationary in G space. You are either moving through 3 space or moving through the proper time dimension or a combination of both.
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Thanks for spotting that while I was still able to edit the post. Fixed now!