rajadhani tatah sabhut
mathura bhagavan yatra
nityam sannihito harih
Translation by His Divine Grace A. C. Bhaktivedanta Swami Srila Prabhupada:
Since that time, the city of Mathura had been the capital of all the kings of the Yadu dynasty. The city and district of Mathura are very intimately connected with Krsna, for Lord Krsna lives there eternally.
Purport by His Divine Grace A. C. Bhaktivedanta Swami Srila Prabhupada:
It is understood that Mathura City is the transcendental abode of Lord Krsna; it is not an ordinary material city, for it is eternally connected with the Supreme Personality of Godhead. Vrndavana is within the jurisdiction of Mathura, and it still continues to exist. Because Mathura and Vrndavana are intimately connected with Krsna eternally, it is said that Lord Krsna never leaves Vrndavana (vrndavanam parityajya padam ekam na gacchati). At present, the place known as Vrndavana, in the district of Mathura, continues its position as a transcendental place, and certainly anyone who goes there becomes transcendentally purified. Navadvipa-dhama is also intimately connected with Vrajabhumi. Srila Narottama dasa Thakura therefore says:
sri gauda-mandala-bhumi, yeba jane cintamani,
tara haya vrajabhume vasa
“Vrajabhumi” refers to Mathura-Vrndavana, and Gauda-mandala-bhumi includes Navadvipa. These two places are nondifferent. Therefore, anyone living in Navadvipa-dhama, knowing Krsna and Sri Caitanya Mahaprabhu to be the same personality, lives in Vrajabhumi, Mathura-Vrndavana. The Lord has made it convenient for the conditioned soul to live in Mathura, Vrndavana and Navadvipa and thus be directly connected with the Supreme Personality of Godhead. Simply by living in these places, one can immediately come in contact with the Lord. There are many devotees who vow never to leave Vrndavana and Mathura. This is undoubtedly a good vow, but if one leaves Vrndavana, Mathura or Navadvipa-dhama for the service of the Lord, he is not disconnected from the Supreme Personality of Godhead. At any rate, we must understand the transcendental importance of Mathura-Vrndavana and Navadvipa-dhama. Anyone who executes devotional service in these places certainly goes back home, back to Godhead, after giving up his body. Thus the words mathura bhagavan yatra nityam sannihito harih are particularly important. A devotee should fully utilize this instruction to the best of his ability. Whenever the Supreme Lord personally appears, He appears in Mathura because of His intimate connection with this place. Therefore although Mathura and Vrndavana are situated on this planet earth, they are transcendental abodes of the Lord.