There are things that one feels like writing about, and things that one does not feel like writing about. And there are things that simply need to be written. Let us examine one such scriptive necessity this balmy evening.
And, as always, let us get right to the point with a fell and generic swoop. One is given to understand that North Indian people disagree with the spellings of certain South Indian names. (Not I, you aver. Okay, not you, but what's a blogpost without sweeping statements?) These Northern individuals cringe when they read supposedly misspelt names such as Geetha and Ananth and, if linguistically-minded, launch into explanations of how ta is different from tha and such. Are they wrong? Not really. Are they right? Not exactly.
Today one shall try, in one's own small way, to lay some of these issues to rest. But before that, let us embark upon a brief phonetics refresher. Let us focus, in particular, on a most marvellous category of utterances: the plosives. As the name suggests, plosives are quite literally little bursts of sound created by your vocal apparatus. The p-, b-, k-, d- and t- sounds are all plosives.
We consider first the retroflex plosives, pronounced by flicking the tongue off the palate. The voiced and voiceless  retroflex plosives are the Hindi  ड and ट respectively. These may be aspirated  to give us ढ and ठ.
We note also the dental plosives, articulated by placing the tongue on the teeth. The voiced and voiceless dental plosives are the Hindi द and त respectively. These may be aspirated to give us ध and थ.
Tamil uses roughly the same sounds, but economises on characters: all of the dental plosives are represented by த் and all of the retroflex plosives by ட், with the exact pronunciation depending on the word (and possibly the accent). Effectively, t and d are not entirely distinct and aspiration is loosely enforced.
But there are also the alveolar plosives, which are articulated midway between the retroflex and dental varieties. Alveolar plosives do not strictly exist in Hindi or Tamil at all, but they are the English t and d. Many Indian speakers of English recognize this: when you say tip, your tongue is closer to your teeth than when you pronounce a rollicking Hindi word like 'tashan'. These alveolar plosives can also, in principle, be either aspirated or unaspirated – but in British English, there is usually an inevitable, if mild, 'default' aspiration. Tip almost becomes ठिप.
We return, now, to the central point: why are Gauthams almost certainly South Indian and Gautams likely North Indian? It's merely a question of transliterative mapping. For some sounds the mapping is straightforward: the letter s was always the obvious choice for the स sound in Hindi and the ஸ் sound in Tamil. But transliterating the t- and d- plosives is more complicated, because English has only the letters t and d to work with  and because these letters do not correspond exactly to any Hindi or Tamil sound.
As a way around this, both Hindi and Tamil transliterators use the 'th' and 'dh' digraphs. Unfortunately, they use these digraphs in different ways.
Hindi transliteration uses 'th' and 'dh' to indicate aspiration. This seems natural to Hindi speakers, but these digraphs are not generally used for that purpose in English: the British th is often a non-plosive lisp, and dh is rare. For the t-sounds, the conventional  Hindi-English mapping is as follows:
Tamil transliteration, by contrast, conventionally uses 'th' and 'dh' to indicate dentality. The accepted Tamil-English mapping for t-sounds is as follows:
It is precisely this difference in mapping that leads to the confusion between our Northern and Southern friends. In particular, the dental t̪ is a 't' under Hindi convention but a 'th' under Tamil convention. Hence, Geetha and Ananth. And Gautham. A sound that is common to Hindi and Tamil can be transliterated differently into English under the different mappings of the two languages.
Which mapping is, phonetically, better? Neither, really. English doesn’t make the dental/retroflex distinction to begin with, and doesn't really make the aspirated/unaspirated distinction either, so ol' Wodehouse might have said something like ʈʰ when faced with either t or th . Transcribing vernacular sounds using English letters is a subjective exercise because English has different (and often fewer) sounds.
So, dear reader, it doesn't matter whether she signs off as Geeta or Geetha; she almost certainly wants to be called gi: t̪ɑ:. And, unless the world graduates to writing everything directly in the International Phonetic Alphabet, that's something we'll have to make our peace with.
 'Voiced' and 'voiceless' refer to whether the vocal cords vibrate or not. If you suspect your vocal cords vibrate for both ट and ड , consider the halant-versions instead.
 The term 'Hindi' is generally used throughout this post, but the same matters apply to Sanskrit and most other North Indian languages.
 Funnily, 'aspiration' refers to inhalation in medicine but to exhalation in phonetics. Deal with it, people.
 Indian speakers of English tend to avoid this aspiration precisely because they distinguish rigidly between aspirated and un-aspirated plosives.
 Some have used capital letters to add another degree of freedom, using t for dental and T for retroflex plosives. While this isn’t a bad idea, it is unheard of in Bollywood. So it does not count.
 Indian English, inter alia, replaces the lisped 'th' of 'think' with the aspirated t̪ʰ / थ
 A case could be made for 'adhere', perhaps.
 And Bollywood-blessed
 Using International Phonetic Alphabet (IPA) notation
 And he would've made a fine joke about it, too.