In regard to Mathurā, we find this quotation:
mathyate tu jagat sarvaṁ
brahma-jñānena yena vā
tat-sāra-bhūtaṁ yad yasyāṁ
mathurā sā nigadyate
When a self-realized soul acts in his transcendental position, his situation is called Mathurā. In other words, when one acts in the process of bhakti-yoga, he may live anywhere, but actually he lives in Mathurā, Vṛndāvana. Devotion to Kṛṣṇa, the son of Nanda Mahārāja, is the essence of all knowledge, and wherever such knowledge is manifested is called Mathurā. Also, when one establishes bhakti-yoga, excluding all other methods, one's situation is called Mathurā. Yatra nityaṁ sannihito hariḥ: the place where Hari, the Supreme Personality of Godhead, lives eternally is called Mathurā. The word nitya indicates eternality. The Supreme Lord is eternal, and His abode is also eternal. Goloka eva nivasaty akhilātma-bhūtaḥ (Bs. 5.37). Although the Lord is always stationed in His abode, Goloka Vṛndāvana, He is present everywhere in fullness. This means that when the Supreme Lord descends on the surface of the world, His original abode is not vacant, for He can remain in His original abode and simultaneously descend upon Mathurā, Vṛndāvana, Ayodhyā and other places. He does not need to descend, since He is already present there; He simply manifests Himself.
(Srimad Bhagavatam--------10:1-------summary).

Since that time, the city of Mathurā had been the capital of all the kings of the Yadu dynasty. The city and district of Mathurāare very intimately connected with Kṛṣṇa, for Lord Kṛṣṇa lives there eternally.
It is understood that Mathurā City is the transcendental abode of Lord Kṛṣṇa; it is not an ordinary material city, for it is eternally connected with the Supreme Personality of Godhead. Vṛndāvana is within the jurisdiction of Mathurā, and it still continues to exist. Because Mathurā and Vṛndāvana are intimately connected with Kṛṣṇa eternally, it is said that Lord Kṛṣṇa never leaves Vṛndāvana (vṛndāvanaṁ parityajya padam ekaṁ na gacchati). At present, the place known as Vṛndāvana, in the district of Mathurā, continues its position as a transcendental place, and certainly anyone who goes there becomes transcendentally purified. Navadvīpa-dhāma is also intimately connected with Vrajabhūmi.
(Srimad Bhagavatam--------10:1:28------translation and purport).

"Vrajabhūmi" refers to Mathurā-Vṛndāvana, and Gauḍa-maṇḍala-bhūmi includes Navadvīpa. These two places are nondifferent. Therefore, anyone living in Navadvīpa-dhāma, knowing Kṛṣṇa and Śrī Caitanya Mahāprabhu to be the same personality, lives in Vrajabhūmi, Mathurā-Vṛndāvana. The Lord has made it convenient for the conditioned soul to live in Mathurā, Vṛndāvana and Navadvīpa 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 Vṛndāvana and Mathurā. This is undoubtedly a good vow, but if one leaves Vṛndāvana, Mathurā or Navadvīpa-dhāma 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 Mathurā-Vṛndāvana and Navadvīpa-dhāma. Anyone who executes devotional service in these places certainly goes back home, back to Godhead, after giving up his body. Thus the words mathurā bhagavān yatra nityaṁ sannihito hariḥ are particularly important. A devotee should fully utilize this instruction to the best of his ability. Whenever the Supreme Lord personally appears, He appears inMathurā because of His intimate connection with this place. Therefore although Mathurā and Vṛndāvana are situated on this planet earth, they are transcendental abodes of the Lord.
(Srimad Bhagavatam--------10:1:28-------purport).

That is the verdict of the Bhagavad-gītā. But those who are in the darkness of ignorance conclude that the Lord's birth and activities in the material world are similar to those of the ordinary living being. Such imperfect conclusions cannot give anyone liberation. His birth, therefore, in the family of King Yadu as the son of King Vasudeva and His transfer into the family of Nanda Mahārāja in the land of Mathurā are all transcendental arrangements made by the internal potency of the Lord. The fortunes of the Yadu dynasty and that of the inhabitants of the land of Mathurā cannot be materially estimated. If simply by knowing the transcendental nature of the birth and activities of the Lord one can get liberation easily, we can just imagine what is in store for those who actually enjoyed the company of the Lord in person as a family member or as a neighbor. All those who were fortunate enough to associate with the Lord, the husband of the goddess of fortune, certainly obtained something more than what is known as liberation. Therefore, rightly, the dynasty and the land are both ever glorious by the grace of the Lord.
(Srimad Bhagavatam--------1:10:26-------purport).

Considering all this, the heavenly planets are more celebrated than the earth. But the celebrity of earth has defeated that of the heavenly planets because of Dvārakā, where Lord Śrī Kṛṣṇa reigned as King. Three places, namely Vṛndāvana, Mathurā and Dvārakā, are more important than the famous planets within the universe. These places are perpetually sanctified because whenever the Lord descends on earth He displays His transcendental activities particularly in these three places. They are perpetually the holy lands of the Lord, and the inhabitants still take advantage of the holy places, even though the Lord is now out of their sight. The Lord is the soul of all living beings, and He desires always to have all the living beings, in their svarūpa, in their constitutional position, to participate in transcendental life in His association. His attractive features and sweet smiles go deep into the heart of everyone, and once it is so done the living being is admitted into the kingdom of God, from which no one returns. This is confirmed in the Bhagavad-gītā.
(Srimad Bhagavatam--------1:10:27-------purport).

Hastināpura, the capital of the Pāṇḍavas, was situated somewhere near present New Delhi, and the kingdom of Ugrasena was situated in Mathurā. While returning to Delhi from Dvārakā, Arjuna must have visited the city of Mathurā, and therefore the inquiry about the King of Mathurā is valid. Amongst various names of the relatives, the name of Rāma or Balarāma, eldest brother of Lord Kṛṣṇa, is added with the words "the Personality of Godhead" because Lord Balarāma is the immediate expansion of viṣṇu-tattva as prakāśa-vigraha of Lord Kṛṣṇa. The Supreme Lord, although one without a second, expands Himself as many other living beings. The viṣṇu-tattva living beings are expansions of the Supreme Lord, and all of them are qualitatively and quantitatively equal with the Lord. But expansions of the jīva-śakti, the category of the ordinary living beings, are not at all equal with the Lord.
(Srimad Bhagavatam--------1:14:28-29-------purport).

These sacred places are called the four dhāmas. Dhāma refers to a place where one can immediately contact the Supreme Lord. To go to Badarī-nārāyaṇa one has to pass through Hardwar on the path to the Supreme Personality of Godhead. Similarly, there are other holy places of pilgrimage, such as Prayāga (Allahabad) and Mathurā, and the topmost of them all is Vṛndāvana. Unless one is very advanced in spiritual life, it is recommended that he live in such holy places and execute devotional service there. But an advanced devotee like Nārada Muni who is engaged in preaching work can serve the Supreme Lord anywhere. Sometimes he even goes to the hellish planets. Hellish conditions do not affect Nārada Muni because he is engaged in greatly responsible activities in devotional service. According to the statement of Nārada Muni, Madhuvana, which is still existing in the Vṛndāvana area, in the district ofMathurā, is a most sacred place. Many saintly persons still live there and engage in the devotional service of the Lord.
(Srimad Bhagavatam--------4:8:42-------purport).

Those who are filled with ecstatic love for Kṛṣṇa always see the form of Śyāmasundara within their hearts. Raghupati Upādhyāya confirms that the Absolute Truth, the Supreme Personality of Godhead, has many incarnations—Nārāyaṇa, Nṛsiṁha, Varāha and others—but Kṛṣṇa is distinguished as the supermost. According to Śrīmad-Bhāgavatam (1.3.28), kṛṣṇas tu bhagavān svayam: "Kṛṣṇa is the original Personality of Godhead." Kṛṣṇa means Śyāmasundara, who plays His flute in Vṛndāvana. Of all forms, this form is the best of all. Kṛṣṇa lives sometimes in Mathurā and sometimes in Dvārakā, but Mathurā is considered the better place. This is also confirmed by Rūpa Gosvāmī in his Upadeśāmṛta (9): vaikuṇṭhāj janito varā madhu-purī. "Madhu-purī, or Mathurā, is far superior to the Vaikuṇṭhalokas in the spiritual world."
(Sri Caitanya Caritamrta-------2:19:102---------purport).

Śrīla Bhaktivinoda Ṭhākura informs us that this letter from Rūpa Gosvāmī to Sanātana Gosvāmī is mentioned by the annotator of the Udbhaṭa-candrikā. Śrīla Rūpa Gosvāmī wrote a note to Sanātana Gosvāmī from Bāklā. This note indicated that Śrī Caitanya Mahāprabhu was coming to Mathurā, and it stated:
yadu-pateḥ kva gatā mathurā-purī
raghu-pateḥ kva gatottara-kośalā
iti vicintya kurusva manaḥ sthiraṁ
na sad idaṁ jagad ity avadhāraya
"Where has the Mathurā-purī of Yadupati gone? Where has the Northern Kośalā of Raghupati gone? By reflection, make the mind steady, thinking, "This universe is not eternal.""
(Sri Caitanya Caritamrta-------2:30:3---------purport).