Ngā Tauira Pātai me te Kōrero mō te Whakawhanaungatanga o Einstein
Ko te whanaungatanga o Einstein tētahi o ngā ariā tino taketake o te ahupūngao hou, e whakarerekē ana i te huarahi e mārama ai tātou ki te wāhi me te wā. E rua ngā wāhanga o tēnei ariā: te whanaungatanga motuhake (1905) me te whanaungatanga whānui (1915). I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira e pā ana ki te whanaungatanga o Einstein, ā, ka matapakihia ēnei hei whakarato i tētahi māramatanga hohonu ake.
Te Whakawhanaungatanga Motuhake
Ko te ariā whanaungatanga motuhake e pā ana ki ngā mea e neke ana i ngā tere pumau e tata ana ki te tere o te mārama. E rua ngā hua matua o tēnei ariā ko te whakawhānui i te wā me te whakapoto i te roa.
1. Te Whakawhanuitanga o te Wā
Mena e rua ngā kaimātaki, tētahi e tū ana i runga i te whenua, tētahi e tere haere ana, ka rerekē ngā wā e inehia ana e rāua mō te kaupapa kotahi.
Tauira raruraru:
E neke ana te kairangi moana i te 0.8 ngā wā o te tere o te mārama (c) ki tētahi whetū e 10 tau-mārama te tawhiti atu i a Papatūānuku. Kia pēhea te roa e tae atu ai te kairangi moana ki te whetū?
Kōrero:
Tuatahi, ka tatauhia e tātou te wā i inehia e te kaimātakitaki i runga i te Ao:
\[ t_B = \frac{d}{v} = \frac{10 \text{ tau mārama}}{0.8 \, c} = 12.5 \text{ tau} \]
Hei tatau i te wā i inehia e te kairangi aorangi (te whakanui i te wā), ka whakamahia e mātou te tātai:
\[ t_A = t_B \sqrt{1 – \frac{v^2}{c^2}} \]
Whakakapia ngā uara e mōhiotia ana:
\[ t_A = 12.5 \sqrt{1 – (0.8)^2} \]
\[ t_A = 12.5 \sqrt{1 – 0.64} \]
\[ t_A = 12.5 \sqrt{0.36} \]
\[ t_A = 12.5 \whakareatia ki te 0.6 \]
\[ t_A = 7.5 \text{ tau} \]
Nō reira, ko te wā i inehia e ngā kairangi moana he 7.5 tau.
2. Ngā Whakaiti Roa
Ina neke tetahi mea i te tere e tata ana ki te tere o te mārama, ka poto ake tōna roa ki te kaimātakitaki tūmau.
Tauira raruraru:
Kei te tere te waka mokowhiti, he 10 mita te roa, ā, e 0.9 ngā wā o te tere o te mārama. Kia pēhea te roa o te waka mokowhiti ki tētahi kaimātakitaki i runga i te Ao?
Kōrero:
Hei tatau i te whakawhāititanga roa, ka whakamahia e mātou te tātai:
\[ L = L_0 \sqrt{1 – \frac{v^2}{c^2}} \]
Kei hea:
– Ko te roa tika, ko te roa tūturu rānei (10 mita),
– Ko te tere o te waka rererangi ko \( v \) (0.9c).
Whakakapia ngā uara e mōhiotia ana:
\[ L = 10 \sqrt{1 – (0.9)^2} \]
\[ L = 10 \sqrt{1 – 0.81} \]
\[ L = 10 \sqrt{0.19} \]
\[ L = 10 \whakareatia ki te 0.436 \]
\[ L = 4.36 \text{ mita} \]
Nō reira, ko te roa o te waka rererangi e ai ki ngā kaimātakitaki i runga i te Papatūānuku he 4.36 mita.
Whānuitanga Whānui
Ka kōrerohia te whanaungatanga whānui mō te kaha ā-papatipu, arā, ko te wāhi me te wā e pāngia ana e te papatipu me te pūngao.
3. Arotahi Ā-Toi
Ka puta te arotahi ā-papatipu ina piko te mārama mai i tētahi mea tawhiti e te kaha ā-papatipu o tētahi mea nui pērā i te tupuni whetu, i te poka pango rānei.
Tauira raruraru:
He nui te taumaha o te tupuni A hei whakakā i te mārama mai i te quasar B, kei muri i a ia. Mena he 1.5 hēkona pewa te koki whakakā, he aha te taumaha o te tupuni A? (Whakamahia te pūmau ā-papatipu a Newton, G = 6.674×10^-11 N(m/kg)^2, te tere o te mārama c = 3×10^8 m/s)
Kōrero:
Ka taea te hoatu i te koki o te whakapēhanga θ mā te tātai:
\[ \theta = \frac{4GM}{c^2 R} \]
Kei hea:
– Ko te pūmau ā-papatipu te \( G \),
– Ko te papatipu o te tupuni ko \( M \)
– Ko te tere o te mārama te \( c \),
– Ko te \( R \) te tawhiti tata rawa atu i waenganui i te mārama me te pokapū o te tupuni.
I te mea e hiahia ana tātou ki te kimi i a M, ka whakarerekētia e tātou te tātai:
\[ M = \frac{\theta c^2 R}{4G} \]
Me kī ko R he 5×10^20 mita (te tawhiti toharite o ngā tupuni whetu). Tahurihia a θ mai i ngā hēkona āwhata ki ngā rātiana (1 hēkona āwhata = 4.848×10^-6 rātiana):
\[ \theta = 1.5 \times 4.848 \times 10^{-6} \, \text{radian} = 7.272 \times 10^{-6} \, \text{radian} \]
Whakakapia ngā uara e mōhiotia ana:
\[ M = \frac{(7.272 \times 10^{-6}) (3 \times 10^8)^2 (5 \times 10^{20})}{4 \times 6.674 \times 10^{-11}} \]
\[ M = \frac{(7.272 \times 10^{-6}) (9 \times 10^{16}) (5 \times 10^{20})}{26.696 \times 10^{-11}} \]
\[ M = \frac{(3.2764 \times 10^{31})}{26.696 \times 10^{-11}} \]
\[ M = 1.227 \times 10^{41} \, \text{kg} \]
Nō reira, ko te papatipu o te tupuni A he tata ki te 1.227×10^41 kirokaramu.
4. Te Whakatairanga o te Perihelion o Mercury
Ka taea hoki e te whanaungatanga whānui te whakamārama i te hekenga o te porowhita o te aorangi a Mercury, he mea kāore e taea te whakamārama e ngā tikanga miihini Newtonian.
Tauira raruraru:
He aha te rahi o te nekehanga perihelion o Mercury e ai ki te whakamāramatanga whānui? (Tauwhātanga whanaungatanga A: 43 arcseconds ia rautau)
Kōrero:
Whakamahia tika ngā raraunga kua whakaratohia:
E ai ki te ariā whānui a Einstein mō te whanaungatanga, ko te nekehanga perihelion kua whakaahuatia o Mercury he 43 hēkona pewa ia rautau, he rite anō hoki ki ngā hua o te tirohanga.
Ngā tauira:
Mā te whakaoti i ēnei tauira raruraru me ngā kōrero, ka taea e tātou te kite me pēhea te whakarato a te whanaungatanga o Einstein i tētahi māramatanga hohonu ake mō te wā, te roa, me te kaha ā-papa. Kāore i te mea i whakarerekē noa tēnei ariā i tā tātou tirohanga pūtaiao mō te ao whānui, engari he tono mahi anō hoki i roto i ngā hangarau hou, pērā i ngā pūnaha whakatere GPS, e hiahia ana ki ngā whakatikatika whanaungatanga kia tika ai te mahi. Ko te ako me te mārama ki te whanaungatanga o Einstein he taahiraa nui ki te ruku hohonu atu ki te ao matatini o te ahupūngao.
