php表单提交错误500

我正试图在我的网站上添加一个带有摘要的简单联系表单，但没有很成功。每当我尝试测试我的表单时，我总是收到一个错误500。该表单指向位于我的服务器上的单独的php文件。

我真的是php的新手，我正在努力找出问题所在。这是表单的html：

<form method="post" action="form.php" id="contact-form" class="col s12">   
    <div class="row">
        <div class="input-field col m6 s12">
            <input id="first_name" type="text" class="validate">
            <label for="first_name">First Name</label>
        </div>
        <div class="input-field col m6 s12">
            <input id="last_name" type="text" class="validate">
            <label for="last_name">Last Name</label>
        </div>
    </div>
    <div class="row">
        <div class="input-field col s12">
            <input id="email" type="email" class="validate"required>
            <label for="email">Email</label>
        </div>
    </div>
    <div class="row">
        <div class="input-field col s12">
            <textarea id="msg" class="materialize-textarea"></textarea>
            <label for="msg">Message</label>
        </div>
    </div>
    <div class="row">
        <div class="col s12 m6">
            <div class="g-recaptcha" data-sitekey="xxxx-Xxxxxxxxxxxxxxx-xxxxxxxxxxxxxxxxxxxx" data-type="image"></div>
        </div>
        <div class="col s12 m6">
            <button class="btn btn-large waves-effect waves-light" type="submit" name="action">Send Message</button>
        </div>
    </div>
</form>

这是form.php:

<?php
$secret='xxx-Xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx';
$sitekey='xxx-Xxxxxxxxxxxxxxx-xxxxxxxxxxxxxxxxxxxx';
$first_name=$last_name=$email=$msg="";
$email = test_input($_POST["email"]);
if (!filter_var($email, FILTER_VALIDATE_EMAIL)) {
      $emailErr = "Invalid email format"; 
}
$msg = test_input($_POST["msg"]);
if (isset($_POST['first_name'])) {
    $first_name=$_POST['first_name'];
    if (!preg_match("/^[a-zA-Z ]*$/",$name)) {
          $nameErr = "Only letters and white space allowed"; 
    }
}
if (isset($_POST['last_name'])) {
    $last_name=$_POST['last_name'];
    if (!preg_match("/^[a-zA-Z ]*$/",$name)) {
          $nameErr = "Only letters and white space allowed"; 
    }
}
$response=file_get_contents("https://www.google.com/recaptcha/api/siteverify?secret=$secret&response=".$captcha);
if ($response.success==false) {
    echo '<p>Captcha failed.<p>'
    exit;
}
$toEmail='contact@example.org';
$message = '<p><b>First Name: </b>'.$first_name.'</p>'.
'<p><b>Last Name: </b>'.$last_name.'</p>'.
'<p><b>Email: </b>'.$email.'</p>'.
'<p><b>Message: </b><br/>'.$msg.'</p>';
$headers = 'From: first last <contact@example.org> '."rn".
'Reply-To: '.$first_name.$last_name.' <'.$email.'> '. "rn".
'Content-type: text/html'. "rn".
'X-Mailer: PHP/'.phpversion();
$message2 = '<p><b>Here is a copy of your message: </b><br/><br/>'.$first_name.' '.$last_name.'wrote: <br/><br/></p>'.'<p>'.$msg'</p>' 
$headers2 = 'From:'.$toEmail."rn".'Reply-To: '.$toEmail."rn".
'Content-type: text/html'. "rn".
'X-Mailer: PHP/'.phpversion();
mail($toEmail,'New Inquery from Website',$message,$headers);
mail($email,'Copy of your form submission',$message2,$headers2);

function test_input($data) {
    $data = trim($data);
    $data = stripslashes($data);
    $data = htmlspecialchars($data);
    return $data;
}
?>

我的php/html有什么问题吗？还是别的什么？我的网络服务器在Ubuntu 14.04上使用php5-fpm和nginx。谢谢你的帮助！