Tambayoyi Misali Game da Tasirin Dangantakar Einstein
Ka'idojin Einstein na dangantaka, gami da ka'idojinsa na musamman da na gabaɗaya na dangantaka, sun kawo sauyi ga fahimtarmu game da sarari, lokaci, da nauyi. Duk da cewa Einstein ya fara gabatar da waɗannan ka'idoji a farkon ƙarni na 20, tasirinsu ga kimiyya da fasaha na zamani ya yi zurfi sosai. Wannan labarin zai bincika misalai da dama na matsaloli waɗanda ke bincika tasirin dangantakar Einstein a cikin yanayi daban-daban kuma ya nuna yadda wannan ka'idar ta canza yanayin kimiyya.
Misali Tambaya ta 1: Faɗaɗa Lokaci da Tafiya a Sararin Samaniya
Tambaya:
Wani ɗan sama jannati yana tafiya zuwa ga tauraro shekaru 4 na haske daga Duniya a ninki 0,8 na saurin haske (0,8c). Har yaushe ɗan sama jannatin yake tunanin tafiyar za ta ɗauka?
Tattaunawa:
Domin fahimtar abin da ke faruwa na faɗaɗa lokaci, muna amfani da dabarar asali ta dangantaka ta musamman:
\[t' = \frac{t}{\gamma} \]
inda \( \gamma \) shine factor na Lorentz wanda aka bayar ta hanyar:
\[ \gamma = \frac{1}{\sqrt{1 – \left(\frac{v}{c}\right)^2}} \]
A nan, \( v = 0,8c \) da \( c \) shine saurin haske. Sannan,
\[ \gamma = \frac{1}{\sqrt{1 – (0,8)^2}} = \frac{1}{\sqrt{1 – 0,64}} = \frac{1}{\sqrt{0,36}} = \frac{1}{0,6} \kimanin 1,667 \]
Idan nisan tauraron shekaru 4 ne na haske kuma ɗan sama jannatin yana tafiya da gudun 0,8c, lokacin da mai lura a Duniya (t) ya gani shine:
\[ t = \frac{Nisa}{Speed} = \frac{4 \text{ shekaru masu haske}}{0,8c} = 5 \text{ shekaru} \]
Duk da haka, lokacin da ɗan sama jannatin (t') ya fuskanta shine:
\[ t' = \frac{t}{\gamma} = \frac{5 \text{ shekaru}}{1,667} \approx 3 \text{ shekaru} \]
Don haka, a cewar 'yan sama jannatin, tafiyar ta ɗauki kimanin shekaru 3 kacal, duk da cewa daga mahangar Duniya ta ɗauki shekaru 5.
Misali Tambaya ta 2: Matsewar Tsawon Lokaci da Lura da Gwaji
Tambaya:
Tsawon jirgin sama yana da tsawon mita 100 idan aka auna shi a hutawa idan aka kwatanta da Duniya. Idan jirgin sama yana tafiya a gudun 0,6c idan aka kwatanta da wanda ke kallo a Duniya, tsawon lokacin da yake ɗauka ga mai kallo a Duniya?
Tattaunawa:
Matsewar tsayi wani tasirin dangantaka ne da aka bayyana ta hanyar dangantaka ta musamman, wanda aka bayyana ta:
\[ L = L_0 \sqrt{1 – \left(\frac{v}{c}\right)^2} \]
inda \( L_0 \) shine tsawon abin da ke hutawa, \( v \) shine saurin dangi, kuma \( L \) shine tsawon abin a saurin dangi. Ga jirgin sama:
\[ L_0 = 100 \rubutu{ mita}, \; v = 0,6c, \rubutu{ sannan} \]
\[ L = L_0 \sqrt{1 – \left(\frac{v}{c}\right)^2} = 100 \sqrt{1 – (0,6)^2} = 100 \sqrt{1 – 0,36} = 100 \sqrt{0,64} = 100 \sqrt{0,8 = 80 \text{mita} \]
Don haka, tsawon jirgin a cewar masu lura da shi a Duniya mita 80 ne.
Misali Tambaya ta 3: Nauyi da Ka'idar Dangantaka ta Gabaɗaya a GPS
Tambaya:
Taurarin GPS suna zagaya Duniya a tsayin kilomita 20.200 sama da saman Duniya a gudun kimanin kilomita 3,874/s. Ta amfani da kwatancen gabaɗaya, ƙididdige gyaran lokaci da tauraron GPS ke buƙatar yi kowace rana don yin la'akari da tasirin nauyi na Duniya.
Tattaunawa:
Tauraron GPS dole ne su daidaita lokacinsu don manyan tasirin guda biyu: faɗaɗa lokaci saboda saurin gudu (dangantaka ta musamman) da faɗaɗa lokaci saboda nauyi (dangantaka ta gabaɗaya). Duk da haka, za mu mayar da hankali kan tasirin nauyi a nan:
Ta amfani da ka'idar dangantaka ta gabaɗaya, lokaci zai wuce a hankali a cikin wani yanki mai ƙarfi na nauyi. Tsarin da ke nuna nauyi a bayyane daga dangantaka ta gabaɗaya shine:
\[ t_g = t_0 \left( 1 – \frac{2GM}{Rc^2} \right) \]
inda \(R \) yake nisan daga tsakiyar nauyi, \(G \) shine ma'aunin nauyi, \(M \) shine nauyin Duniya, \(c \) shine saurin haske, kuma \(t_0 \) shine lokacin mai lura da 'tsayawa' a saman Duniya.
An bayar:
– Tashin Duniya, \( M \kimanin 5,972 \sau 10^{24} \text{ kg} \)
– Radius na Duniya, \( R_{\text{surface}} \approx 6.371 \times 10^6 \text{ m} \)
– Tsayin tauraron dan adam, \( H = 20.200 \sau 10^3 \rubutu{ m} \)
– Don haka nisan daga tsakiyar Duniya zuwa tauraron dan adam, \( R = R_{\text{surface}} + H \approx 26.571 \times 10^6 \text{ m} \)
Bambancin lokaci a kowace rana tsakanin tauraron dan adam da saman Duniya, idan aka yi la'akari da nauyi kawai:
\[ \Delta t_g \approx \frac{2GM}{c^2} \left( \frac{1}{R_{\text{surface}}} – \frac{1}{R} \right) \]
Maye gurbinsa:
\[ \Delta t_g \approx \frac{2 \times 6,67408 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2} \times 5,972 \times 10^{24} \text{ kg}}{(3 \times 10^8 \text{ m/s})^2} \left( \frac{1}{6,371 \times 10^6 \text{ m}} – \frac{1}{26,571 \times 10^6 \text{ m}} \right) \]
Bayan lissafi, wannan sakamakon yana daidai da gyaran lokaci na yau da kullun ga tauraron GPS, wanda yake da ƙarancin microseconds fiye da lokacin saman Duniya. Saboda haka, tauraron GPS suna buƙatar yin la'akari da wannan tasirin don kiyaye daidaito.
Babban Tasiri Kan Fasaha da Fahimtar Duniya
Waɗannan misalan sun bayyana karara cewa dangantakar Einstein ba wai kawai ka'idar zahiri ce kawai ba, har ma tana da fa'idodi masu yawa na aiki. Daga faɗaɗa lokaci a cikin tafiye-tafiyen sararin samaniya zuwa raguwar tsayi da gyaran lokaci a fasahar GPS, dangantakar Einstein ta yi tasiri mai mahimmanci.
Sabbin abubuwa a fannoni daban-daban na fasaha, kimiyya, har ma da falsafa suna nuna tasirin ka'idar. Dangantaka ta ba da damar zurfafa bincike kan sararin samaniya, haɓaka fasahar sadarwa mai ci gaba, da sabbin fahimtar ilimin sararin samaniya da ramukan baƙi.
A ƙarshe, ka'idar Einstein ta dangantaka ta ci gaba da kasancewa muhimmin ɓangare na nazarin kimiyyar lissafi ta zamani kuma tana ci gaba da zama tushen wahayi da bincike ga masana kimiyya a duk duniya.
