Here we will investigate the apparent curvature caused by space and time squaring to different signs.
In space we can identify a point by 3 scalar values: (x,y,z). Similarly in space-time we can identify a point (representing an event say) by 4 scalar values: (t,x,y,z). So is space-time just a matter of increasing our rectangular coordinate system from 3 dimensions to 4?
There is another difference, in space we would represent the distance between two points as:
distance = √(Δx²+Δy²+Δz²)
Which means that points of equal distance from the origin (we will use the origin to represent the observer as the origin has no special significance in euclidean space) can be represented as spheres:
However in space-time then we either use a metric where (-+++):
- space squares to positive
- time squares to negative
- space squares to negative
- time squares to positive
The second seems to be more common in physics literature but I don't think it makes much difference at this stage so I think I'll use the first alterative for now to make it easier to compare with the way that we usually measure things in space. So we will represent the distance between two points as:
distance = √(-Δt²+Δx²+Δy²+Δz²)
This change of sign caused by time squaring to negative causes a big difference when plotting points of equal distance from the origin which now become parabolas:
So in space-time we can still plot t,x,y,z on a rectangular coordinate system but we have to be careful about its meaning, we can't draw any conclusions from the proximity of points in space to their closeness by the measure that we have chosen.
The way we have chosen to measure the concept of 'distance' between points then goes on to affect most other concepts in geometry. For instance, the definition of a 'straight line' is usually the shortest distance between two points, so should we define a straight line in space-time in terms of minimising this distance function √(-Δt²+Δx²+Δy²+Δz²). Although changing the definition of distance could affect what is a straight line does it in this instance? After all we still experience light rays as traveling in straight lines in 3D space (until we go on to general relativity).
To reinforce the strangeness of defining distance in this way lets look at some of the consequences:
The outer cone shown above (the null cone: two cones meeting at the same point) shows all the points that are a zero distance from the origin (where Δt²=Δx²+Δy²+Δz²). However although they are all a zero distance from the origin they are not all zero distance from each other!
The distances to points outside the null cone are imaginary (Δt²>Δx²+Δy²+Δz² so distance=√-ve ) indicating that we would have to go faster than the speed of light to reach these points.
So not only is this not a Euclidean space or a vector space (a=b and b=c does not imply a=c) it is very hard to imagine there could be a transform that would map this onto a Euclidean space.
How can we analyse such a situation? I think we have to treat the origin as the position of the observer and accept that the space will appear different to different observers.
Conjugate Space and Invariance
In Euclidean space the distance from the origin to the end of the vector (a scalar value) is given by:
d = √(v•v)
- v = vector
- d = distance (scalar)
Although we have used the dot product we could also have used the geometric product (since the cross product with itself is zero: v×v=0).
On this page we are working in Minkowski space so we need to modify the above expression to give our definition of distance.
So in addition to the space-time definition:
v = (t,x,y,z)
We also need to define its conjugate:
v' = (-t,x,y,z)
We can therefore express the distance scalar as follows:
d = √(v•v')
Hapa tutachunguza mwinuko dhahiri unasababishwa na nafasi na muda squaring kwa ishara tofauti. Katika nafasi tunaweza kutambua uhakika na 3 scalar maadili: (x,y,z). Kwa mfano, katika nafasi-wakati tunaweza kutambua hatua (kuwakilisha tukio kusema) kwa 4 scalar maadili: (t, x, y, z). Kwa hiyo, wakati na nafasi ni suala la kuongeza mfumo wetu wa kuratibu wa mstatili kutoka vipimo vitatu hadi vinne. "Kama ilivyoelezwa hapo juu, ""mzuri"" ni ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa na ""mzuri"" kwa sababu ni sawa." Kwa hiyo, tunaonyesha umbali kati ya pointi mbili kama: umbali = − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Njia tumechagua kupima dhana ya'muda' kati ya pointi kisha inaendelea kuathiri dhana nyingine nyingi katika jiometri. Kwa mfano, ufafanuzi wa'mstari wa moja kwa moja' ni kawaida umbali mfupi kati ya pointi mbili, hivyo tunapaswa kufafanua mstari wa moja kwa moja katika nafasi-wakati katika suala la kupunguza umbali huu kazi <unk> (-Δt2+Δx2+Δy2+Δz2). Ingawa kubadilisha ufafanuzi wa umbali inaweza kuathiri nini ni mstari wa moja kwa moja haina katika kesi hii? Kwa kweli, tunatambua kwamba miale ya mwanga husafiri kwa njia ya mstari wa moja kwa moja katika nafasi ya 3D, lakini ni kwa sababu ya nadharia ya upendeleo. Kwa kuimarisha ajabu ya kufafanua umbali kwa njia hii hebu kuangalia baadhi ya matokeo: nje koni iliyoonyeshwa hapo juu (koni null: koni mbili kukutana katika hatua moja) inaonyesha pointi zote ambazo ni umbali sifuri kutoka chanzo (ambapo Δt2 = Δx2 + Δy2 + Δz2). Hata hivyo ingawa wote ni umbali sifuri kutoka chanzo wao si wote umbali sifuri kutoka kwa kila mmoja! "Mafarasi ya vituo nje ya koni ya null ni ""wa kuwaziwa"" (Dt2=Dx2+Dy2+Dz2 = distance=√-ve), ikionyesha kwamba tunapaswa kwenda kwa kasi kuliko mwendo wa mwanga ili kufikia vituo hivyo." Kwa hiyo, si tu kwamba hii si nafasi ya Euclidean au nafasi ya vector (a = b na b = c haina maana ya a = c) ni vigumu sana kufikiria kuna inaweza kuwa kubadilisha kwamba ramani hii juu ya nafasi ya Euclidean. Twaweza kuchanganuaje hali kama hiyo? Nadhani tunapaswa kutibu chanzo kama nafasi ya mtazamaji na kukubali kwamba nafasi itaonekana tofauti kwa watazamaji tofauti. Katika nafasi ya Euclidean, umbali kutoka mwanzo hadi mwisho wa vector (thamani ya scalar) hutolewa na: d = v = vector - d = umbali (scalar) Ingawa tumetumia bidhaa ya dot, tunaweza pia kutumia bidhaa ya kijiometri (kwa kuwa bidhaa ya msalaba na yenyewe ni sifuri: v × v = 0). Katika ukurasa huu sisi ni kufanya kazi katika nafasi Minkowski hivyo tunahitaji kurekebisha usemi wa juu kutoa ufafanuzi wetu wa umbali. Kwa hiyo, kwa kuongezea ufafanuzi wa nafasi-wakati: v = (t, x, y, z) Tunahitaji pia kufafanua conjugate yake: v'= (-t, x, y, z) Kwa hivyo tunaweza kuelezea scalar ya umbali kama ifuatavyo: d = √ (v) v '