<div dir="ltr"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><span style="font-family:arial,sans-serif">On Wed, Aug 16, 2017 at 7:25 AM, Stuart LaForge </span><span dir="ltr" style="font-family:arial,sans-serif"><<a href="mailto:avant@sollegro.com" target="_blank">avant@sollegro.com</a>></span><span style="font-family:arial,sans-serif"> wrote:</span><br></div><div class="gmail_extra"><div class="gmail_quote"><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-left:1ex"><i><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">> </div>What fraction of reality, by 4-D volume, lies inside<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>of our past lightcone?</i></blockquote><div><br></div><div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"></div><font size="4">We know the Big Bang happened 13.8 billion years ago but it seems to me that to answer your question I'd have to know how big our future lightcone is, and I don't know what that is<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">, for all I know it's infinite.</div><br></font></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-left:1ex"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">><i> </i></div><i>I modelled reality as a 4-D hypersphere or 4-sphere<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>in Planck units given by<br>
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<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>S4. t^2+x^2+y^2+z^2 <= R^2<br></i></blockquote><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-left:1ex"><i> </i></blockquote><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-left:1ex"><i><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div> where t is time and R is the radius of the 4-sphere. </i></blockquote><div> </div><div><br></div><div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"></div><font size="4">The<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>Pythagoras Theorem<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>works for 4-D Euclidian space but that's not what Einstein used, he used 4-D<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>Minkowski<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>space (sometimes called<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>Hyperbolic<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>space)<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>with a different distance formula<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>than the one Pythagoras gave:</font></div><div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"><font size="4"><br></font></div></div><font size="4">R^2= X^2 +Y^2 +Z^2 - (c^2)T^2</font><br><div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"><br></div></div><font size="4"><div class="gmail_default" style="display:inline">Where</div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"></div> R is the distance between events in spacetime<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> and</div> X,Y, and Z are the distance in space in those dimensions and (c^2)T^2 is the distance in the time dimension. The c is the speed of light and the c^2 is a conversion factor<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> to get the units right;</div> if you didn't have that you'd be adding apples and oranges, or rather meters and seconds<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">,</div> which would make no sense.<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div></font></div><div class="gmail_quote"><font size="4"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"><br></div></font></div><div class="gmail_quote"><font size="4"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"></div></font><font size="4"><div class="gmail_default" style="display:inline"><font face="arial, helvetica, sans-serif"></font></div></font>I<font size="4">t can be proven that<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>in<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>flat space mathematically there are only 2 possible definitions of distance such that distance remains<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>the same for all observers<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>in any frame of reference, the one in<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>the<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>Pythagoras Theorem<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>and the one above. Both are equally valid mathematically but<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>Minkowski<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>works better for physical reasons. Minkowski treats time differently than space and that's why the minus sign is in there, if it were a plus as in regular old 4-D Euclidian space then causality would not be<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>preserved<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>and a event could happen before the thing that caused it, and that would be unphysical. And the above formula is only an approximation, when Spacetime becomes highly curved, as it is around massive stars,<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>4-D<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>Tensor Calculus must be used<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>to find the distance between two events in spacetime.</font></div><div class="gmail_quote"><font size="4"><br></font></div><div class="gmail_quote"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"><font size="4"> John K Clark</font></div></div><div class="gmail_quote"><div><br></div><div> </div></div></div></div>