Ngā Tauira Pātai e Matapaki ana i te Pānga o te Whakawhanaungatanga a Einstein
Kua hurihia e ngā ariā a Einstein mō te whanaungatanga, tae atu ki āna ariā motuhake me ngā ariā whānui mō te whanaungatanga, tō tātou māramatanga ki te wāhi, te wā, me te kaha ā-papa. Ahakoa i whakaurua tuatahitia ēnei ariā e Einstein i te tīmatanga o te rautau 20, he nui te pānga ki te pūtaiao me te hangarau hou. Ka tirohia e tēnei tuhinga ētahi tauira raruraru e tūhura ana i te pānga nui o te whanaungatanga a Einstein i roto i ngā horopaki rerekē, ā, ka whakaatu i te āhua o te whakarerekētanga o tēnei ariā i tō tātou tauira pūtaiao.
Tauira Pātai 1: Te Whakawhanuitanga o te Wā me te Haereere i te Wāhi
Pātai:
E haere ana tētahi kairangi moana ki tētahi whetū, e 4 tau mārama te tawhiti atu i te Ao, he 0,8 ngā wā tere atu i te tere o te mārama (0,8c). E whakaaro ana te kairangi moana kia pēhea te roa o te haerenga?
Kōrero:
Hei mārama ki te āhuatanga o te whakawhānui i te wā, ka whakamahia e mātou te tauira taketake o te whanaungatanga motuhake:
\[ t' = \frac{t}{\gamma} \]
ko \( \gamma \) te tauwehenga Lorentz e homai ana e:
\[ \gamma = \frac{1}{\sqrt{1 – \left(\frac{v}{c}\right)^2}} \]
I konei, ko \( v = 0,8c \) me \( c \) te tere o te mārama. Kātahi,
\[ \gamma = \frac{1}{\sqrt{1 – (0,8)^2}} = \frac{1}{\sqrt{1 – 0,64}} = \frac{1}{\sqrt{0,36}} = \frac{1}{0,6} \approx 1,667 \]
Mena he 4 tau mārama te tawhiti o te whetū, ā, e tere ana te kairangi i te 0,8c, ko te wā i kitea e te kaimātakitaki i runga i te Ao (t) ko:
\[ t = \frac{Tawhiti}{Tere} = \frac{4 \text{ tau mārama}}{0,8c} = 5 \text{ tau} \]
Heoi, ko te wā i pā ki te kairangi moana (t') ko:
\[ t' = \frac{t}{\gamma} = \frac{5 \text{ tau}}{1,667} \approx 3 \text{ tau} \]
Nō reira, e ai ki ngā kairangi moana, e toru tau noa iho te roa o te haerenga, ahakoa e rima tau te roa mai i te tirohanga a Papatūānuku.
Tauira Pātai 2: Te Whakaiti Roa me te Tirotiro Whakamātautau
Pātai:
E 100 mita te roa o tētahi waka mokowhiti i te wā e takoto ana, e pā ana ki a Papatūānuku. Mēnā kei te neke te waka mokowhiti i te tere o te 0,6c e pā ana ki tētahi kaimātakitaki i runga i te Papatūānuku, e hia te roa o tōna āhua ki tētahi kaimātakitaki i runga i te Papatūānuku?
Kōrero:
Ko te whakapoto roa tētahi atu pānga whanaungatanga e whakaahuatia ana e te whanaungatanga motuhake, e whakaatuhia ana e:
\[ L = L_0 \sqrt{1 – \left(\frac{v}{c}\right)^2} \]
ko \( L_0 \) te roa o te mea e okioki ana, ko \( v \) te tere whanaunga, ā, ko \( L \) te roa o te mea i te tere whanaunga. Mō te papa:
\[ L_0 = 100 \text{ mita}, \; v = 0,6c, \text{ kātahi} \]
\[ L = L_0 \sqrt{1 – \left(\frac{v}{c}\right)^2} = 100 \sqrt{1 – (0,6)^2} = 100 \sqrt{1 – 0,36} = 100 \sqrt{0,64} = 100 \times 0,8 = 80 \text{ mita} \]
Nō reira, e ai ki ngā kaimātakitaki o te Ao, ko te roa o te waka rererangi he 80 mita.
Tauira Pātai 3: Te Āhuatanga Papatipu me te Ariā o te Whakawhanaungatanga Whānui i roto i te GPS
Pātai:
Ka porowhita ngā amiorangi GPS i te Ao i te teitei o te 20.200 kiromita i runga ake i te mata o te Ao i te tere o te 3,874 kiromita/hēkona. Mā te whakamahi i te whanaungatanga whānui, tatauhia te whakatikatika wā e hiahiatia ana e ngā amiorangi GPS i ia rā hei whakaaro i ngā pānga o te kaha ā-papa o te Ao.
Kōrero:
Me whakarerekē e ngā amiorangi GPS ō rātou wā mō ngā pānga matua e rua: te whakawhanuitanga o te wā nā ngā tere teitei (te whanaungatanga motuhake) me te whakawhanuitanga o te wā nā te kaha ā-papatipu (te whanaungatanga whānui). Heoi, ka arotahi tātou ki te pānga o te kaha ā-papatipu i konei:
Mā te whakamahi i te ariā o te whanaungatanga whānui, ka pōturi ake te haere o te wā i roto i te papa kaha ake. Ko te tātai mō te kaha kitea mai i te whanaungatanga whānui ko:
\[ t_g = t_0 \left( 1 – \frac{2GM}{Rc^2} \right) \]
ko \( R \) te tawhiti mai i te pokapū o te kaha ā-papatipu, ko \( G \) te pūmau kaha ā-papatipu, ko \( M \) te papatipu o te Ao, ko \( c \) te tere o te mārama, ā, ko \( t_0 \) te wā o tētahi kaimātakitaki 'tūturu' i runga i te mata o te Ao.
I hoatu:
– Te Taumaha o te Ao, \( M \approx 5,972 \times 10^{24} \text{ kg} \)
– Te pūtoro o te Ao, \( R_{\text{surface}} \approx 6.371 \times 10^6 \text{ m} \)
– Teitei o te amiorangi, \( H = 20.200 \times 10^3 \text{ m} \)
– Nō reira, ko te tawhiti mai i te pokapū o te Ao ki te amiorangi, \( R = R_{\text{surface}} + H \approx 26.571 \times 10^6 \text{ m} \)
Ko te rerekētanga wā i ia rā i waenga i te amiorangi me te mata o te Ao, me te whakaaro noa ki te kaha ā-papa:
\[ \Delta t_g \approx \frac{2GM}{c^2} \left( \frac{1}{R_{\text{surface}}} – \frac{1}{R} \right) \]
Te whakakapi:
\[ \Delta t_g \approx \frac{2 \times 6,67408 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2} \times 5,972 \times 10^{24} \text{ kg}}{(3 \times 10^8 \text{ m/s})^2} \left( \frac{1}{6,371 \times 10^6 \text{ m}} – \frac{1}{26,571 \times 10^6 \text{ m}} \right) \]
I muri i te tatau, ka rite tēnei hua ki te whakatikatika wā o ia rā mō ngā amiorangi GPS, he 7 maikorohekona te puhoi ake i te wā o te mata o te Ao. Nō reira, me whai whakaaro ngā amiorangi GPS ki tēnei pānga kia mau tonu ai te tika.
Te Pānga Nui ki te Hangarau me te Māramatanga ki te Ao Whānui
Mā ēnei tauira e mārama ai ehara te ariā ā-tinana o Einstein i te ariā ā-tinana noa iho, engari he whānui hoki ōna tono mahi. Mai i te whakawhanuitanga o te wā i roto i te haere ki te wāhi ki te whakapoto i te roa me te whakatikatika wā i roto i te hangarau GPS, he nui te pānga o te ariā ā-tinana o Einstein.
Ko ngā auahatanga i roto i ngā momo mara hangarau, pūtaiao, tae noa ki te rapunga whakaaro e whakaatu ana i te awe o te ariā. Nā te whanaungatanga i taea ai te tūhura hohonu ake i te wāhi, te whanaketanga o ngā hangarau whakawhitiwhiti kōrero matatau ake, me ngā māramatanga hou mō te ao tūroa me ngā pōro pango.
Hei whakamutunga, ko te ariā whanaungatanga a Einstein kei te noho tonu hei wāhanga nui o te ako i te ahupūngao hou, ā, kei te noho tonu hei puna whakahihiko me te tūhuratanga mō ngā kaipūtaiao puta noa i te ao.
